Skip to main content
Log in

Topological quantum field theory and invariants of graphs for quantum groups

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

On the basis of generalized 6j-symbols we give a formulation of topological quantum field theories for 3-manifolds including observables in the form of coloured graphs. It is shown that the 6j-symbols associated with deformations of the classical groups at primitive even roots of unity provide examples of this construction. Calculational methods are developed which, in particular, yield the dimensions of the state spaces as well as a rather simple proof of the relation, previously found by Turaev and Walker for the case ofU q (sl(2,C)), between these models and corresponding ones based on the ribbon graph construction of Reshetikhin and Turaev.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • [A] Andersen, H.H.: Tensor products of quantized tilting modules. Commun. math. Phys.149, 149 (1992)

    Google Scholar 

  • [At] Atiyah, M.: Topological quantum field theories. Publ. Math. I.H.E.S.68, 175 (1989)

    Google Scholar 

  • [D] Durhuus, B.: A discrete approach to topological quantum field theories. J. Geom. and Phys.11, 155 (1993)

    Google Scholar 

  • [DJN] Durhuus, B., Jakobsen, H., Nest, R.: Topological quantum field theories from generalized 6j-symbols. Rev. Math. Phys.5, 1 (1993)

    Google Scholar 

  • [Dr] Drinfeld, V.G.: Quantum groups. Proc. of ICM Berkeley 1986. Providence, R.I.,1 (1987)

  • [FG] Felder, G., Grandjean, O.: On combinatorial three-manifold invariant. Preprint (1992)

  • [KMS] Karowski, M., Müller, W., Schrader, R.: State sum invariants of compact 3-manifolds with boundary and 6j-symbols. J. Phys. A25, 4847 (1992)

    Google Scholar 

  • [KS] Karowski, M., Schrader, R.: A combinatorial approach to topological quantum field theories and invariants of graphs. Commun. Math. Phys.151, 355 (1993)

    Google Scholar 

  • [KS1] Karowski, M., Schrader R.: Private communication

  • [N] Nill, F.: Private communication

  • [RT1] Reshetikhin, N., Turaev, V.: Ribbon graphs and their invariants derived from quantum groups. Commun. Math. Phys.127, 26 (1990)

    Google Scholar 

  • [RT2] Reshetikhin, N., Turaev, V.: Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math.103, 547 (1991)

    Google Scholar 

  • [T] Turaev, V.: Quantum invariants of 3-manifolds I. Preprint 509/p-295, Strasbourg, (1992) to be published in Quantum invariants of 3-manifolds, Berlin: Walter de Gruyter, 1994

  • [T1] Turaev, V.: Topology of shadows. Preprint (1991)

  • [T2] Turaev, V.: Quantum invariants of links and 3-valent graphs in 3-manifolds. Publ. IHES 77, 121 (1993)

    Google Scholar 

  • [TV] Turaev, V., Viro, O.: State sum invariants of 3-manifolds and quantum 6j-symbols. Topology31, 865 (1992)

    Google Scholar 

  • [TW] Turaev, V., Wenzl, H.: Quantum invariants of 3-manifolds associated with classical simple Lie algebras. Int. J. Math.4, 323 (1991)

    Google Scholar 

  • [V] Verlinde, E.: Fusion rules and modular transformations in 2d conformal field theory. Nucl. Phys.B300, 360 (1988)

    Google Scholar 

  • [Wa] Walker, K.: On Witten's 3-manifold invariants. Preprint (1991)

  • [Wi1] Witten, E.: Topological quantum field theory. Commun. Math. Phys.117, 353 (1988)

    Google Scholar 

  • [Wi2] Witten, E.: Quantum field theory and the Jones polynomial Commun. Math. Phys.121, 351 (1989)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Communicated by G. Felder

Supported by DAAD and DFG, SFB 288 “Differentialgeometrie und Quantenphysik”

Rights and permissions

Reprints and permissions

About this article

Cite this article

Beliakova, A., Durhuus, B. Topological quantum field theory and invariants of graphs for quantum groups. Commun.Math. Phys. 167, 395–429 (1995). https://doi.org/10.1007/BF02100592

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02100592

Keywords

Navigation