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Generalized Homologies for the Zero Modes of the SU(2) WZNW Model

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Abstract

We generalize the BRS method for the (finite-dimensional) quantum gauge theory involved in the zero modes of the monodromy extended SU(2) WZNW model. The generalization consists of a nilpotent operator Q such that Qh = 0 (h = k + 2 = 2, 3, ... being the height of the current algebra representation) acting on an extended state space. The physical subquotient is identified with the direct sum ⊕ h-1 n=1 Ker(Qn)/Im(Qh−n).

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References

  1. Alekseev, A. Yu. and Faddeev, L. D.: (T*G)t: A toy model for conformal field theory, Comm. Math. Phys. 141 (1991), 413–422.

    Google Scholar 

  2. Becchi, C., Rouet, A. and Stora, R.: Renormalization models with broken symmetries, In: G. Velo and A. S. Wightman (eds), Renormalization Theory (Erice 1975), D. Reidel, Dordrecht, 1976.

    Google Scholar 

  3. Bernard, D. and Felder, G.: Fock representations and BRST cohomology in SL.2. current algebra, Comm. Math. Phys. 127 (1990), 145–168.

    Google Scholar 

  4. Bonora, L. and Cotta-Ramusino, P.: Some remarks on BRS transformations, anomalies and the cohomology of the Lie algebra of the group of gauge transformations, Comm. Math. Phys. 87 (1983), 589–603.

    Google Scholar 

  5. Dubois-Violette, M.: Systèmes dynamiques contraints: l'approche homologique, Ann. Inst. Fourier Grenoble 37 (1987), 45–57.

    Google Scholar 

  6. Dubois-Violette, M.: Generalized differential spaces with d N = 0 and the q-differential calculus. Czech J. Phys. 46 (1997), 1227–1233.

    Google Scholar 

  7. Dubois-Violette, M.: d N = 0: Generalized homology, K-Theory 14 (1998), 371–404.

    Google Scholar 

  8. Dubois-Violette M.: Generalized homologies for d N = 0 and graded q-differential algebras, Contemp. Math. 219 (1998), 69–79.

    Google Scholar 

  9. Dubois-Violette, M. and Kerner, R.: Universal q-differential calculus and q-analog of homological algebra, Acta Math. Univ. Comenian. 65 (1996), 175–188.

    Google Scholar 

  10. Dubois-Violette, M. and Todorov, I. T.: Generalized cohomologies and the physical subspace of the SU.2. WZNW model, Lett. Math. Phys. 42 (1997), 183–192.

    Google Scholar 

  11. Faddeev, L. D.: On the exchange matrix for a WZNW model, Comm. Math. Phys. 132 (1990), 131–138.; Quantum symmetry in conformal field theory by Hamiltonian methods, In: J. Fröhlich et al. (eds), New Symmetry Principles in Quantum Field Theory, Plenum, New York, 1992, pp. 159–175.

    Google Scholar 

  12. Falceto, F. and Gawedzki, K.: Lattice Wess-Zumino-Witten model and quantum groups, J. Geom. Phys. 11 (1993), 251–279.

    Google Scholar 

  13. Furlan, P., Hadjiivanov, L. K. and Todorov, I. T.: Canonical approach to the quantum WZNW model, Trieste-Vienna preprint IC/95/74; ESI 234 (1995).

  14. Furlan, P., Hadjiivanov, L. K. and Todorov, I. T.: Operator realization of the SU(2) WZNW model. hep-th/9602101; Nuclear Phys. B 474 (1996), 497–511.

    Google Scholar 

  15. Furlan, P., Hadjiivanov, L. K. and Todorov, I. T.: A quantum gauge group approach to the 2D SU(n) WZNW model. hep-th/9610202; Internat. J. Modern Phys. A 12 (1997), 23–32.

    Google Scholar 

  16. Gawedzki, K.: Classical origin of quantum group symmetries in Wess-Zumino-Witten conformal field theory, Comm. Math. Phys. 139 (1991), 201–213.

    Google Scholar 

  17. Greub, W. H.: Linear Algebra, Springer-Verlag, New York, 1963.

    Google Scholar 

  18. Guichardet, A.: Cohomologie des groupes topologiques et des algèbres de Lie, CEDIC/Fernand Nathan, Paris, 1980.

    Google Scholar 

  19. Hadjiivanov, L. K., Isaev, A. P., Ogievetsky, O. V., Pyatov, P. N. and Todorov, I. T.: Hecke algebraic properties of dynamical R-matrices, Applications to related quantum matrix algebras. q-alg/9712026; J. Math. Phys. 40 (1999), 427–448.

    Google Scholar 

  20. Kapranov, M. M.: On the q-analog of homological algebra, Preprint, Cornell University 1991; q-alg/9611005.

  21. Kassel, C. and Wambst, M.: Algèbre homologique des N-complexes et homologies de Hochschild aux racines de l'unité, Publ. RIMS Kyoto Univ. 34 (1998), 91–114.

    Google Scholar 

  22. Knizhnik, V. G. and Zamolodchikov, A. B.: Current algebra and Wess-Zumino model in two dimensions, Nuclear Phys. B 247 (1984), 83–103.

    Google Scholar 

  23. Loday, J.-L.: Cyclic homology, Springer-Verlag, New York, 1992.

    Google Scholar 

  24. Sullivan, D.: Infinitesimal computations in topology. Publ. IHES 47 (1977), 269–331.

    Google Scholar 

  25. Witten, E.: Non-abelian bosonization in two dimensions, Comm.Math. Phys. 92 (1984), 455–472.

    Google Scholar 

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Authors and Affiliations

  1. Laboratoire de Physique Théorique, Université Paris XI, Bâtiment 210, F-91405, Orsay Cedex, France

    Michel Dubois-Violette

  2. Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, BG-1784, Sofia, Bulgaria

    Ivan T. Todorov

Authors
  1. Michel Dubois-Violette
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  2. Ivan T. Todorov
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Dubois-Violette, M., Todorov, I.T. Generalized Homologies for the Zero Modes of the SU(2) WZNW Model. Letters in Mathematical Physics 48, 323–338 (1999). https://doi.org/10.1023/A:1007679216588

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