Communications in Mathematical Physics

, Volume 172, Issue 2, pp 317–358 | Cite as

Combinatorial quantization of the Hamiltonian Chern-Simons theory I

  • Anton Yu. Alekseev
  • Harald Grosse
  • Volker Schomerus


Motivated by a recent paper of Fock and Rosly [6] we describe a mathematically precise quantization of the Hamiltonian Chern-Simons theory. We introduce the Chern-Simons theory on the lattice which is expected to reproduce the results of the continuous theory exactly. The lattice model enjoys the symmetry with respect to a quantum gauge group. Using this fact we construct the algebra of observables of the Hamiltonian Chern-Simons theory equipped with a *- operation and a positive inner product.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Gauge Group 
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  1. 1.
    Mack, G., Schomerus, V.: Action of truncated quantum groups on quasi quantum planes and a quasi-associative differential geometry and calculus. Commun. Math. Phys.149, 513 (1992)Google Scholar
  2. 2.
    Mack, G., Schomerus, V.: Quasi Hopf quantum symmetry in quantum theory. Nucl. Phys.B370, 185 (1992)Google Scholar
  3. 3.
    Sudbury, A.: Non-commuting coordinates and differential operators. In: Quantum groups, T. Curtright et al. (eds), Singapore: World Scientific, (1991)Google Scholar
  4. 4.
    Kirillov, A.N., Reshetikhin, N.: Representations of the algebraU q(sl(2)), q-orthogonal polynomials and invariants of links. Preprint LOMI E-9-88, Leningrad (1988)Google Scholar
  5. 5.
    Boulatov, D.V.: q-deformed lattice gauge theory and three manifold invariants. Int. J. Mod. Phys.A8, 3139 (1993)Google Scholar
  6. 6.
    Fock, V.V., Rosly, A.A.: Poisson structures on moduli of flat connections on Riemann surfaces andr-matrices. Preprint ITEP 72-92, June 1992, MoscowGoogle Scholar
  7. 7.
    Schomerus, V.: Quantum symmetry in quantum theory. PhD thesis, DESY-93-18Google Scholar
  8. 8.
    Schomerus, V.: Construction of field algebras with quantum symmetry from local observables. Commun. Math. Phys., to appearGoogle Scholar
  9. 9.
    Schomerus, V.: Quasi quantum group covariant q-oscillators. Nucl. Phys.B401, 455 (1993)Google Scholar
  10. 10.
    Drinfel'd, V.G.: Quantum groups. Proc. ICM 798 (1987)Google Scholar
  11. 11.
    Drinfel'd, V.G.: Quasi Hopf algebras and Knizhnik Zamolodchikov equations. In: Problems of modern quantum field theory, Proceedings Alushta 1989, Research Reports in Physics, Heidelberg: Springer Verlag 1989; Drinfel'd, V.G.: Quasi-Hopf algebras. Leningrad. Math. J. Vol.1, No. 6 (1990)Google Scholar
  12. 12.
    Scheunert, M.: The antipode of and the star operation in a Hopf algebra. J. Math. Phys.34, 320 (1993)Google Scholar
  13. 13.
    Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys.121, 351 (1989)Google Scholar
  14. 14.
    Elitzur, S., Moore, G., Schwimmer, A., Seiberg, N.: Remarks on the canonical quantization of the Chern-Simons-Witten theory. Nucl. Phys.326, 108 (1989)Google Scholar
  15. 15.
    Alekseev, A., Faddeev, L., Shatashvili, S.: Quantization of symplectic orbits of compact Lie groups by means of the functional integral. Geom. and Phys.5, No. 3, 391–406 (1989)Google Scholar
  16. 16.
    Axelrod, S., Singer, I.M.: Chern-Simons Perturbation Theory II. Preprint HEP-TH/9304087Google Scholar
  17. 17.
    Gerasimov, A.: Localization in GWZW and Verlinde formula. Uppsala preprint HEP-TH/9305090Google Scholar
  18. 18.
    Piunikhin, S.: Combinatorial expression for universal Vassiliev link invariant. Harvard preprint HEP-TH/9302084Google Scholar
  19. 19.
    Alekseev, A.Yu., Faddeev, L.D., Semenov-Tian-Shansky, M.A.: Hidden quantum groups inside Kac-Moody algebras. Commun. Math. Phys.149, No. 2, 335 (1992)Google Scholar
  20. 20.
    Bar-Natan, D., Witten, E.: Perturbative Expansion of Chern-Simons Theory with Non-compact Gauge Group. Commun. Math. Phys.141, 423 (1991)Google Scholar
  21. 21.
    Alekseev, A.Yu.: Integrability in the Hamiltonian Chern-Simons theory. Uppsala preprint HEP-TH/9311074, to appear in St.-Petersburg. Math. JGoogle Scholar
  22. 22.
    Altschuler, D., Coste, A.: Quasi-quantum groups, knots, three manifolds and topological field theory. Commun. Math. Phys.150, 83 (1992)Google Scholar
  23. 23.
    Cartier, P.: Construction Combinatorire des Invariants de Vassiliev-Kont-sevich des Noeds. Preprint of the Erwin Schrödinger Institute ESI 15 (1993), Vienna, AustriaGoogle Scholar
  24. 24.
    Turaev, V.G., Viro, O.Y.: State sum of 3-manifolds and quantum 6j-symbols. LOMI preprint (1990)Google Scholar
  25. 25.
    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
  26. 26.
    Karowski, M., Schrader, R.: A combinatorial approach to topological quantum field theories and invariants of graphs. Commun. Math. Phys.151, 355 (1993); Karowski, M., Schrader, R.: A quantum group version of quantum gauge theories in two dimensions. J. Phys.A 25, L1151 (1992)Google Scholar
  27. 27.
    Witten, E.: On quantum gauge theories in two dimensions. Commun. Math. Phys.141, 153 (1991)Google Scholar
  28. 28.
    Reshetikhin, N., Turaev, V.G.: Ribbon graphs and their invariants derived from quantum groups. Commun. Math. Phys.127, 1 (1990)Google Scholar
  29. 29.
    Fröhlich, J., King, C.: The Chern-Simons theory and knot polynomials. Commun. Math. Phys.126, 167 (1989)Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Anton Yu. Alekseev
    • 1
  • Harald Grosse
    • 2
  • Volker Schomerus
    • 3
  1. 1.Institute of Theoretical PhysicsUppsala UniversityUppsalaSweden
  2. 2.Institut für Theoretische PhysikUniversität WienAustria
  3. 3.Department of PhysicsHarvard UniversityCambridgeUSA

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