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SIGMA 4 (2008), 078, 30 pages arXiv:0801.1445
https://doi.org/10.3842/SIGMA.2008.078
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
Abstract
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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References
- 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.
- Hagen C.R.,
A new gauge theory without an elementary photon,
Ann. Physics 157 (1984), 342-359.
- Polyakov A.M., Fermi-Bose transmutations induced by gauge fields,
Modern Phys. Lett. A 3 (1988), 325-328.
- Witten E., Quantum field theory and the Jones polynomial,
Comm. Math. Phys. 121 (1989), 351-399.
- 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.
- 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.
- Deligne P., Théorie de Hodge. II,
Inst. Hautes Études Sci. Publ. Math. (1971), no. 40, 5-58.
- Beilinson A.A., Higher regulators and values of L-functions,
J. Soviet Math. 30 (1985), 2036-2070.
- 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.
- 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.
- Brylinski J.L.,
Loop spaces, characteristic classes and geometric quantization,
Progress in Mathematics, Vol. 107, Birkhäuser Boston, Inc., Boston, MA, 1993.
- 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).
- 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.
- 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.
- 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.
- 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.
- Alvarez O., Topological quantization and cohomology,
Comm. Math. Phys. 100 (1985), 279-309.
- 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.
- Witten E., Topological quantum field theory,
Comm. Math. Phys. 117 (1988), 353-386.
- 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.
- 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.
- 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.
- Hopkins M.J., Singer I.M., Quadratic functions in geometry, topology, and M-theory,
J. Differential Geom. 70 (2005), 329-452, math.AT/0211216.
- Woodhouse N.M.J., Geometric quantization, 2nd ed.,
Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1992.
- 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.
- Mackaay M., Picken R.,
Holonomy and parallel transport for Abelian gerbes,
Adv. Math. 170 (2002), 287-339, math.DG/0007053.
- 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).
- Bott R., Tu L.W.,
Differential forms in algebraic topology,
Graduate Texts in Mathematics, Vol. 82, Springer-Verlag, New York - Berlin, 1982.
- Rolfsen D.,
Knots and links, Mathematics Lecture Series, no. 7, Publish or Perish, Inc., Berkeley, Calif., 1976.
- Calugareanu G., Sur les classes d'isotopie des noeuds tridimensionnels et leurs invariants,
Czechoslovak Math. J. 11 (1961), 588-625.
- 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.
- Guadagnini E., Martellini M., Mintchev M., Wilson lines in Chern-Simons theory and link invariants,
Nuclear Phys. B 330 (1990), 575-607.
- 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.
- Feynman R.P., Hibbs A.R.,
Quantum mechanics and path integrals, McGraw-Hill, New York, 1965.
- Coleman S.,
Aspects of symmetry, Cambridge University Press, New York, 1985.
- Elworthy D. and Truman A.,
Feynman maps, Cameron-Martin formulae and anharmonic oscillators,
Ann. Inst. Henri Poincaré Phys. Théor. 41 (1984), 115-142.
- 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.
- Baez J.C.,
Link invariants, holonomy algebras, and functional integration,
J. Funct. Anal. 127 (1995), 108-131, hep-th/9301063.
- Lickorish W.B.R., Invariants for 3-manifolds from the combinatorics of the Jones polynomial,
Pacific J. Math. 149 (1991), 337-386.
- Morton H.R., Strickland P.M., Satellites and surgery invariants, in Knots 90 (Osaka, 1990), Editor A. Kawauchi, de Gruyter, Berlin, 1992.
- Kirby R., A calculus for framed links in S3,
Invent. Math. 45 (1978), 35-56.
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