Advertisement

Communications in Mathematical Physics

, Volume 151, Issue 2, pp 355–402 | Cite as

A combinatorial approach to topological quantum field theories and invariants of graphs

  • M. Karowski
  • R. Schrader
Article

Abstract

The combinatorial state sum of Turaev and Viro for a compact 3-manifold in terms of quantum 6j-symbols is generalized by introducing observables in the form of coloured graphs. They satisfy braiding relations and allow for surgeries and a discussion of cobordism theory. Application of these techniques give the dimension and an explicit basis for the vector space of the topological quantum field theory associated to any Riemann surface with arbitrary coloured punctures.

Keywords

Colour Neural Network Statistical Physic Vector Space Complex System 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Al] Alexander, J.W.: The combinatorial theory of complexes. Ann. Math.31, 294–322 (1930)Google Scholar
  2. [At1] Atiyah, M.: Topological quantum field theories. Publ. Math. Inst. Hautes Etudes Sci. Paris68, 175–186 (1989)Google Scholar
  3. [At2] Atiyah, M.: The geometry and physics of knots. Cambridge: Cambridge Univ. Press 1990Google Scholar
  4. [AC1] Altschuler, D., Coste, A.: Invariant of three-manifolds from finite groups. CERN preprint TH 6204/91 to appear in Proc. XXth Int. Conf. on differential geometric methods in theoretical physics, Singapore: World Scientific, 1991Google Scholar
  5. [AC2] Altschuler, D., Coste, A.: Quasi-quantum groups, knots, three-manifolds, and topological field theory. Commun. Math. Phys., to appearGoogle Scholar
  6. [Bo] Bott, R.: On E. Verlinde's formula in the context of stable bundles. Int. J. Mod. Phys.6, 2847 (1991)Google Scholar
  7. [D] Drinfeld, V.G.: Quasi Hopf algebras and Knizhnik Zamolodchikov equations. In: Belavin, A.A., Klimyk, A.U., Zamolodchikov, A.B. (eds.) Problems of modern quantum field theory. Berlin, Heidelberg, New York: Springer 1989Google Scholar
  8. [DJN1] Durhuus, B., Jakobsen, H.P., Nest, R.: Topological quantum field theories from generalized 6j-symbols. University of Copenhagen, preprint (1991)Google Scholar
  9. [DJN2] Durhuus, B., Jakobsen, H.P., Nest, R.: A construction of topological quantum field theories from 6j-symbols. Nucl. Phys. B (Proc. Suppl.)666 (1991)Google Scholar
  10. [DPR] Dijkgraaf, R., Pasquier, V., Roche, P.: In Proc. Workshop on Integrable Systems and Quantum Groups. Pavia 1990, and in Proc. Int. Coll. on Modern Quantum Field Theory, Tata Institute, Bombay, 1990Google Scholar
  11. [DW] Dijkgraaf, R., Witten, E.: Topological gauge theories and group cohomology. Commun. Math. Phys.129, 393–429 (1990)Google Scholar
  12. [FG] Freed, D., Gompf, R.E.: Computer calculation of Witten's 3-manifold invariant. Commun. Math. Phys.141, 79–117 (1991)Google Scholar
  13. [FRS] Fredenhagen, K., Rehren, K.-H., Schroer, B.: Superselection sectors with braid group statistics and exchange algebras. II. In preparationGoogle Scholar
  14. [FQ] Freed, D., Quinn, F.: Chern-Simons theory with finite gauge groups, preprint, 1991Google Scholar
  15. [GW] Gepner, D., Witten, E.: String theory on group manifolds, Nucl. Phys. B278, 493–549 (1986)Google Scholar
  16. [K] Kirillov, A.N.: Zap. nauch. Semin. LOMI, 1988, v. 168Google Scholar
  17. [KMS] Karowski, M., Müller, W., Schrader, R.: State sum invariants of compact 3-manifolds with boundary and 6j-symbols. J. Phys. A., to appearGoogle Scholar
  18. [KR] Kirillov, A.N., Reshetikhin, N.Yu.: Representation of the algebraU q(sl(2)),q-orthogonal polynomials and invariants of links. In: Kohno, T. (ed.) New developments in the theory of knots. Advanced series in Mathematical Physics Vol. 11. Singapore: World Scientific, 1989Google Scholar
  19. [MS] Moore, G., Seiberg, N.: Polynomial equations for rational conformal field theories. Phys. Lett. B212, 451–460 (1988)Google Scholar
  20. [OS] Osterwalder, K., Schrader, R.: Axioms for euclidean Green's functions. I, II. Commun. Math. Phys.31, 83–112 (1973),42, 281–305 (1975)Google Scholar
  21. [RT1] Reshetikhin, N.Yu., Turaev, V.G.: Ribbon graphs and their invariants derived from quantum groups. Commun. Math. Phys.127, 1–26 (1990)Google Scholar
  22. [RT2] Reshetikhin, N.Yu., Turaev, V.G.: Invariants of 3-manifolds via link polynomials and quantum groups. Inv. Math.103, 547–597 (1991)Google Scholar
  23. [Sp] Spanier, E.H.: Algebraic topology. New York: MacGraw Hill, 1966Google Scholar
  24. [T1] Turaev, V.G.: Quantum invariants of 3-manifolds and a glimpse of shadow topology. Preprint 1990Google Scholar
  25. [T2] Turaev, V.G.: Quantum invariants of links and 3-valent graphs in 3-manifolds. Preprint 1990Google Scholar
  26. [T3] Turaev, V.G.: State sums in low-dimensional topology. Preprint, 1990Google Scholar
  27. [Th] Thaddeus, M.: Conformal field theory and the cohomology of the moduli space of stable bundles. Oxford University preprintGoogle Scholar
  28. [TV] Turaev, V.G., Viro, O.Y.: State sum of 3-manifolds and quantum 6j-symbols. LOMI preprint, to appear in TopologyGoogle Scholar
  29. [Vi1] Vinberg, E.B.: Discrete groups generated by reflections in Lobačevskii spaces. Math. USSR-Sbornik, Vol. I, No. 3, 429–444 (1967)Google Scholar
  30. [Vi2] Vinberg, E.B.: Some examples of crystallographic groups in Lobačevskii spaces. Math. USSR-Sbornik, Vol.7, No. 4, 617–622 (1969)Google Scholar
  31. [Ve] Verlinde, E.: Fusion rules and modular transformations in 2d conformal field theory. Nucl. Phys. B300, 360–376 (1988)Google Scholar
  32. [Wi1] Witten, E.: Topological quantum field theory. Commun. Math. Phys.117, 353–386 (1988)Google Scholar
  33. [Wi2] Witten, E.: Quantum field theory and the Jones polynomials. Commun. Math. Phys.121, 351–399 (1989)Google Scholar
  34. [Wi3] Witten, E.: On quantum gauge theories in two dimensions. Commun. Math. Phys.141, 153–209 (1991)Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • M. Karowski
    • 1
  • R. Schrader
    • 1
  1. 1.Institut für Theoretische PhysikFreie Universität BerlinBerlin 33Germany

Personalised recommendations