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Communications in Mathematical Physics

, Volume 256, Issue 3, pp 513–537 | Cite as

Wilson Surfaces and Higher Dimensional Knot Invariants

  • Alberto S. CattaneoEmail author
  • Carlo A. Rossi
Article

Abstract

An observable for nonabelian, higher-dimensional forms is introduced, its properties are discussed and its expectation value in BF theory is described. This is shown to produce potential and genuine invariants of higher-dimensional knots.

Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Batalin, I.A., Vilkovisky, G.A.: Relativistic S-matrix of dynamical systems with boson and fermion constraints. Phys. Lett. 69 B, 309–312 (1977); Fradkin, E.S., Fradkina, T.E.: Quantization of relativistic systems with boson and fermion first- and second-class constraints. Phys. Lett. 72 B, 343–348 (1978)Google Scholar
  2. 2.
    Bott, R.: Configuration spaces and imbedding invariants. In: Proceedings of the 4th Gökova Geometry–Topology Conference. Tr. J. Math. 20, 1–17 (1996)Google Scholar
  3. 3.
    Bott, R., Taubes, C.: On the self-linking of knots. J. Math. Phys. 35, 5247–5287 (1994)CrossRefGoogle Scholar
  4. 4.
    Cattaneo, A.S., Cotta-Ramusino, P., Fucito, F., Martellini, M., Rinaldi, M., Tanzini, A., Zeni, M.: Four-dimensional Yang–Mills theory as a deformation of topological BF theory. Commun. Math. Phys. 197, 571–621 (1998)Google Scholar
  5. 5.
    Cattaneo, A.S.: P. Cotta-Ramusino and R. Longoni, Configuration spaces and Vassiliev classes in any dimension. Algebra. Geom. Topol. 2, 949–1000 (2002)Google Scholar
  6. 6.
    Cattaneo, A.S., Cotta-Ramusino, P., Rinaldi, M.: Loop and path spaces and four-dimensional BF theories: connections, holonomies and observables. Commun. Math. Phys. 204, 493–524 (1999)Google Scholar
  7. 7.
    Cattaneo, A.S., Cotta-Ramusino, P., Rossi, C.A.: Loop observables for BF theories in any dimension and the cohomology of knots. Lett. Math. Phys. 51, 301–316 (2000)Google Scholar
  8. 8.
    Cattaneo, A.S., Fröhlich, J., Pedrini, B.: Topological field theory interpretation of string topology. Commun. Math. Phys. 240, 397–421 (2003)Google Scholar
  9. 9.
    Cattaneo, A.S., Rossi, C.A.: Higher-dimensional BF theories in the Batalin–Vilkovisky formalism: the BV action and generalized Wilson loops. Commun. Math. Phys. 221, 591–657 (2001)Google Scholar
  10. 10.
    Cattaneo, A.S., Rossi, C.A.: Configuration space invariants of higher dimensional knots. In preparationGoogle Scholar
  11. 11.
    Chas, M., Sullivan, D.: String topology. http://arxiv.org/list/math/9911159, 1999
  12. 12.
    Fock, V.V., Nekrasov, A.A., Rosly and Selivanov, K.G., what we think about the higher dimensional Cheran-Simons theories, ITEP-91-70, Jul 1991; Sakharov Conf. 465–472, (1991)Google Scholar
  13. 13.
    Greub, W., Halperin, S., Vanstone, R.: Connections, Curvature and Cohomology. Vol. II: Lie Groups, Principal Bundles, Characteristic Classes, Pure and Applied Mathematics 47 II, New York–London: Academic Press, 1973Google Scholar
  14. 14.
    Kontsevich, M.: Feynman diagrams and low-dimensional topology. First European Congress of Mathematics, Paris 1992, Volume II, Progress in Mathematics 120, Basel: Birkhäuser, 1994, pp. 97–121Google Scholar
  15. 15.
    Rossi, C.: Invariants of Higher-Dimensional Knots and Topological Quantum Field Theories, Ph. D. thesis, Zurich University 2002, http://www.math.unizh.ch/~asc/RTH.ps
  16. 16.
    Schwarz, A.S.: The partition function of degenerate quadratic functionals and Ray–Singer invariants. Lett. Math. Phys. 2, 247–252 (1978)Google Scholar
  17. 17.
    Segal, G.: Topological structures in string theory. Phil. Trans. R. Soc. Lond. A 359, 1389–1398 (2001)Google Scholar
  18. 18.
    ’t Hooft, G.: On the phase transition towards permanent quark confinement. Nucl. Phys. B 138, 1 (1978); A property of electric and magnetic flux in nonabelian gauge theories. Nucl. Phys. B 153, 141 (1979)Google Scholar
  19. 19.
    Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121, 351–399 (1989)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Institut für MathematikUniversität Zürich–IrchelZürichSwitzerland
  2. 2.D-MATHETH-ZentrumZürichSwitzerland

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