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Mathematical Aspects of Quantum Field Theories pp 233–271Cite as

Deligne-Beilinson Cohomology in U(1) Chern-Simons Theories

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Abstract

In the early years of the 19th Century, after having studied the trajectories of some celestial objects as Asteroid Ceres or Comet Biela, Carl Freidrich Gauss set forth his famous formula for the linking number, thus providing one of the first mathematical tools allowing a characterization of celestial orbits configurations.

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Notes

  1. 1.

    The \(U(1)\) gauge fields used by physicists are related to the top component of the corresponding DB 1-cocycles according to \(\mathcal{A}_\alpha = 2 \pi (\hbar c / e) A_\alpha \), where \(e\) is the charge of the electron, \(\hbar \) is the Planck constant and \(c\) is the speed of light; the field strength tensor is then \(\mathcal{F} = 2 \pi (\hbar c / e) \mathbf{F}\).

  2. 2.

    which in \(\check{\text{ C }}\)ech cohomology is the equivalent of the exterior product.

  3. 3.

    The Chern-Simons action can be generalized to \((4l+3)\)-dimensional manifolds as it is the only dimension where the DB square \(\bar{\mathbf{A}} \star \bar{\mathbf{A}}\) is not zero, \(\bar{\mathbf{A}}\) being a \((2l+1)\)-connection.

  4. 4.

    Strictly speaking there is also a factor \((e/2 \pi \hbar c)^2\) in front of the abelian and non-abelian lagrangians for the reason explained in the first footnote.

  5. 5.

    The first component of this DB cocycle defines a closed \(1\)-current which doesn’t have integral periods, these periods being defined as intersections with \(\varSigma / 2k\).

  6. 6.

    This class rather belongs to the translation group \(\varOmega ^2_{\mathbb {Z}}(M)^* \).

References

  1. C.F. Gauss, Werke, Fünfter Band, Zweiter Abdruck Herausgegeben von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 1877 (http://archive.org/details/Werkecarlf05gausrich)

  2. M. Epple, Orbits of Asteroids, A Braid, and the First Link Invariant, The Mathematical Intelligencer Volume 20 (Springer, Berlin, 1998), pp. 45–52. (Number 1)

    Google Scholar 

  3. H. Hopf, Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche. Math. Ann. 104, 637–665 (1931)

    Article  MathSciNet  Google Scholar 

  4. J.H. Whitehead, An Expression of Hopf’s Invariant as an Integral. Proc. Nat. Acad. Sci. U.S.A. 33(5), 117–123 (1947)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. H.K. Moffat, The degree of knottedness of tangled vortex lines. J. Fluid Mech. 35, 117–129 (1969)

    Article  ADS  Google Scholar 

  6. L. Woltjer, On hydromagnetic equilibrium. Proc. Nat. Acad. Sci. U.S.A. 44(9), 833–841 (1958)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. S.-S. Chern, J. Simons, Some cohomology classes in principal fiber bundles and their application to Riemannian geometry. Proc. Nat. Acad. Sci. 68(4), 791–794 (1971)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. P. Deligne, Théorie de Hodge. II. Inst. Hautes Études Sci. Publ. Math. 40, 5–58 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  9. A.A. Beilinson, Higher regulators and values of \(L\)-functions. J. Sov. Math. 30, 2036–2070 (1985)

    Article  MATH  Google Scholar 

  10. J. Cheeger, J. Simons, Differential characters and geometric invariants, Stony Brook Preprint (1973)

    Google Scholar 

  11. R. Harvey, B. Lawson, J. Zweck, The de Rham-Federer theory of differential characters and character duality. Am. J. Math. 125, 791–847 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. M.J. Hopkins, I.M. Singer, Quadratic functions in geometry, topology, and M-theory. J. Diff. Geom. 70, 329–452 (2005)

    MathSciNet  MATH  Google Scholar 

  13. J. Simons, D. Sullivan, Axiomatic characterization of ordinary differential cohomology. J. Topol. 1(1), 45–56 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. H. Gillet, Riemann-Roch theorems for higher algebraic K-theory. Adv. Math. 40, 203–289 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  15. H. Esnault, E. Viehweg, Deligne-Beilinson cohomology, in Beilinson’s Conjectures on Special Values of \(L\)-Functions, in Perspectives in Mathematics, vol. 4, ed. by M. Rapaport, P. Schneider, N. Schappacher (Academic Press, Boston, 1988), pp. 43–91

    Google Scholar 

  16. U. Jannsen, Deligne homology, Hodge-\(D\)-conjecture, and motives, in Beilinson’s Conjectures on Special Values of \(L\)-Functions, in Perspectives in Mathematics, vol. 4, ed. by M. Rapaport, P. Schneider, N. Schappacher (Academic Press, Boston, MA, 1988), pp. 305–372

    Google Scholar 

  17. H. Esnault, Recent developments on characteristic classes of flat bundles on complex algebraic manifolds. Jber. d. Dt. Math.-Ver. 98, 182–191 (1996)

    MathSciNet  MATH  Google Scholar 

  18. C. Soulé, Classes caractéristiques secondaires des fibrés plats, Séminaire Bourbaki, 38, Exposé No. 819, p. 14, (1995–1996)

    Google Scholar 

  19. M. Karoubi, Classes caractéristiques de fibrés feuilletés, holomorphes ou algébriques. K-Theory 8, 153–211 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  20. P. Gajer, Geometry of deligne cohomology. Inventiones Mathematicae 127(1), 155–207 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  21. M. Mackaay, R. Picken, Holonomy and parallel transport for Abelian gerbes. Adv. Math. 170, 287–339 (2002). math.DG/0007053

    Article  MathSciNet  MATH  Google Scholar 

  22. O. Alvarez, Topological quantization and cohomology. Commun. Math. Phys. 100, 279 (1985)

    Article  ADS  MATH  Google Scholar 

  23. K. Gawedzki, Topological actions in two-dimensional quantum field theories, in CargÃs̈e 1987, Proceedings of Nonperturbative Quantum Field Theory, pp. 101–141 (1987)

    Google Scholar 

  24. D.S. Freed, Locality and integration in topological field theory, published in Group Theoretical methods in Physics, vol. 2, ed. by M.A. del Olmo, M. Santander, J.M. Guilarte, CIEMAT, pp. 35–54 (1993)

    Google Scholar 

  25. P. Deligne, D. Freed, Quantum fields and strings: a course for mathematicians, 1999 pp. 218–220, vol. 1. ed. by P. Deligne et al., Providence, USA: AMS

    Google Scholar 

  26. J.L. Brylinski, Loop Spaces, Characteristic Classes and Geometric Quantization, Progress in Mathematics, vol. 107 (Birkhäuser Boston Inc, Boston, 1993)

    Book  Google Scholar 

  27. M. Bauer, G. Girardi, R. Stora, F. Thuillier, A class of topological actions, J. High Energy Phys. No. 8, 027, p. 35, hep-th/0406221

  28. E. Guadagnini, F. Thuillier, Deligne-Beilinson cohomology and abelian link invariants. SIGMA 4, 078 (2008)

    MathSciNet  Google Scholar 

  29. F. Thuillier, Deligne-Beilinson cohomology and abelian link invariants: torsion case. J. Math. Phys. 50, 122301 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  30. L. Gallot, E. Pilon, F. Thuillier, Higher dimensional abelian Chern-Simons theories and their link invariants. J. Math. Phys. 54, 022305 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  31. E. Guadagnini, F. Thuillier, Three-manifold invariant from functional integration. J. Math. Phys. 54, 082302 (2013). arXiv:1301.6407

    Article  ADS  MathSciNet  Google Scholar 

  32. W. Ehrenbergand, R.E. Siday, The refractive index in electron optics and the principles of dynamics. Proc. Phys. Soc.B 62, 8–21 (1949)

    Article  ADS  Google Scholar 

  33. Y. Aharonov, D. Bohm, Significance of electromagnetic potentials in quantum theory. Phys. Rev. 115, 485–491 (1959)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. N.M.J. Woodhouse, Geometric Quantization (Clarendon Press, Oxford, 1991)

    Google Scholar 

  35. D. Rolfsen, Knots and Links, Mathematics Lecture Series, no. 7 (Publish or Perish Inc, Berkeley, 1976)

    MATH  Google Scholar 

  36. A.S. Schwarz, The partition function of degenerate quadratic functional and Ray-Singer invariants. Lett. Math. Phys. 2, 247–252 (1978)

    Article  ADS  MATH  Google Scholar 

  37. C.R. Hagen, A new gauge theory without an elementary photon. Ann. Phys. 157, 342–359 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  38. V.F.R. Jones, A polynomial invariant for knots via von Neumann algebras. Bull. Am. Math. Soc. (N.S.) 12, 103–111 (1985)

    Article  MATH  Google Scholar 

  39. E. Witten, Quantum field theory and the Jones polynomial. Comm. Math. Phys. 121, 351–399 (1989)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  40. N.Y. Reshetikhin, V.G. Turaev, Ribbon graphs and their invariants derived from quantum groups. Comm. Math. Phys. 127, 1–26 (1990)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  41. E. Guadagnini, M. Martellini, M. Mintchev, Wilson lines in Chern-Simons theory and link invariants. Nucl. Phys. B 330, 575–607 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  42. H.R. Morton, P.M. Strickland, Satellites and surgery invariants, in Knots 90 (Osaka, 1990), ed. by A. Kawauchi, de Gruyter (Berlin, 1992)

    Google Scholar 

  43. E. Guadagnini, 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)

    Google Scholar 

  44. D. Bar-Natan, Perturbative Chern-Simons Theory. J. Knot Theory Ram. 4–4, 503 (1995)

    Article  MathSciNet  Google Scholar 

  45. A. Hahn, The wilson loop observables of Chern-Simons theory on \(\mathbb{R}^3\) in axial gauge. Comm. Math. Phys. 248, 467–499 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  46. R. Bott, L.W. Tu, Differential Forms in Algebraic Topology (Springer, New York, 1982)

    Book  MATH  Google Scholar 

  47. J. Calais, éléments de théorie des groupes (Presses Universitaires de France, Mathématiques, 1984)

    MATH  Google Scholar 

  48. A. Gramain, Formes d’intersection et d’enlacement sur une variété. Mémoires de la Société Mathématique de France 48, 11–19 (1976)

    MathSciNet  MATH  Google Scholar 

  49. F. Deloup, V. Turaev, On reciprocity. J. Pure Appl. Algebra 208, 153 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  50. C.T.C. Wall, Quadratic forms on finite groups and related topics. Topology 2, 281–298 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  51. E. Guadagnini, F. Thuillier, Path-integral invariants in abelian Chern-Simons theory, Nuclear Physics B (2014), http://dx.doi.org/10.1016/j.nuclphysb.2014.03.009

  52. D. Diakonov, V. Petrov, Non-abelian stokes theorem and quark-monopole interaction. Phys. Lett. B224, 131 (1989)

    Article  ADS  Google Scholar 

  53. C. Beasley, E. Witten, Non-abelian localization for Chern-Simons theory. J. Differ. Geom. 70, 183–323 (2005)

    MathSciNet  MATH  Google Scholar 

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  1. LAPTh, Université de Savoie, CNRS, 9, Chemin de Bellevue, BP 110, 74941, Annecy-le-Vieux Cedex, France

    Frank Thuillier

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  1. Université Montpellier 2, Montpellier, France

    Damien Calaque

  2. Université Claude Bernard Lyon 1, Villeurbanne Cedex, France

    Thomas Strobl

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Thuillier, F. (2015). Deligne-Beilinson Cohomology in U(1) Chern-Simons Theories. In: Calaque, D., Strobl, T. (eds) Mathematical Aspects of Quantum Field Theories. Mathematical Physics Studies. Springer, Cham. https://doi.org/10.1007/978-3-319-09949-1_8

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