Abstract
Two-dimensional, unitary rational conformal field theory is studied from the point of view of the representation theory of chiral algebras. Chiral algebras are equipped with a family of co-multiplications which serve to define tensor product representations. Chiral vertices arise as Clebsch-Gordan operators from tensor product representations to irreducible subrepresentations of a chiral algebra. The algebra of chiral vertices is studied and shown to give rise to representations of the braid groups determined by Yang-Baxter (braid) matrices. Chiral fusion is analyzed. It is shown that the braid- and fusion matrices determine invariants of knots and links. Connections between the representation theories of chiral algebras and of quantum groups are sketched. Finally, it is shown how the local fields of a conformal field theory can be reconstructed from the chiral vertices of two chiral algebras.
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Felder, G., Fröhlich, J., Keller, G.: On the structure of unitary conformal field theory I: Existence of conformal blocks, ETH-preprint November 1988
Zamolodchikov, A.B.: Infinite additional symmetries in two-dimensional conformal quantum field theory. Theor. Math. Phys.65, 1205 (1986)
Verlinde, E.: Fusion rules and modular transformations in 2D conformal field theory. Nucl. Phys. B300, 360 (1988)
Vafa, C.: Toward Classification of conformal field theories. Harvard preprint HUTP-88/A011
Moore, G., Seiberg, N.: Polynomial equations for rational conformal field theories. Phys. Lett. B212, 451 (1988);
—: Naturality in conformal field theory. Nucl. Phys. B313, 16 (1989);
Moore, G., Seiberg, N.: Classical and quantum conformal field theory. IAS preprint IASSNS-HEP-88/39
Felder, G.: BRST approach to minimal models, ETH-preprint August 1988;
Felder, G., Fröhlich, J., Keller, G.: Braid matrices and structure constants for minimal conformal models. IAS preprint 1989
Knizhnik, V.G., Zamolodchikov, A.B.: Current algebra and Wess-Zumino model in two dimensions. Nucl. Phys. B247, 83 (1984);
Gepner, D., Witten, E.: String theory on group manifolds. Nucl. Phys. B278, 493 (1986)
Fröhlich, J.: Statistics of fields, the Yang-Baxter-Equation and the theory of knots and links, Cargèse Lectures 1987. 't Hooft, G. et al. (ed.). New York: Plenum Press 1988
—: Statistics and monodromy in two-dimensional quantum field theory, in “Differential geometric methods in theoretical physics.” Bleuler, K., Werner M. (eds.). Dordrecht, Boston, London: Kluwer 1988
Tsuchiya, A., Kanie, Y.: Vertex operators in the conformal field theory of ℙ1 and monodromy representations of the braid group. Lett. Math. Phys.13, 303 (1987)
Birman, J.: Braids, links and mapping class groups. Ann. Math. Studies, vol.82. Princeton, NJ: Princeton University Press 1974
Goodman, F., de la Harpe, P., Jones, V.: Dynkin diagrams and towers of algebras. Preprint Université de Genève, June 1986
Fröhlich, J., King, C.: Two-dimensional conformal field theory and three-dimensional topology, preprint ETH-TH/89-9
Reshetikhin, N.Yu.: Quantized universal enveloping algebras, the Yang-Baxter equation and invariants of links I, II, LOMI preprint E-4-87, E-17-87
Jones, V.: Hecke algebra representations of braid groups and link polynomials. Ann. Math.126, 335 (1987)
Cappelli, A., Itzykson, C., Zuber, J.B.: Modular invariant partition functions. Nucl. Phys. B280, 445 (1987)
—: The A-D-E Classification of minimal andA (1)1 conformal invariant theories. Commun. Math. Phys.113, 1 (1987)
Dotsenko, V.S., Fateev, V.A.: Conformal algebra and multipoint correlation functions in 2D statistical models. Nucl. Phys. B240, 312 (1984); Four-point correlation functions and operator algebra in 2D conformal invariant theories with central charge ≦1. Nucl. Phys. B251, 691 (1985); Operator algebra of two-dimensional conformal theories with central charge ≦1. Phys. Lett. B154, 291 (1985)
Kohno, T.: Hecke algebra representations of braid groups and classical Yang-Baxter equations, preprint; Linear representations of braid groups and classical Yang-Baxter equations, Contemp. Math., vol.78 (1988), “Braids, Santa Cruz, 1986,” 339
Fateev, V.A., Zamolodchikov, A.B.: Operator algebra and correlation functions in the two-dimensionalsu(2)×su(2) chiral Wess-Zumino model. Sov. J. Nucl. Phys.43, 657 (1986)
Belavin, A.A., Polyakov, A.M., Zamolodchikov, A.B.: Infinite conformal symmetry in two-dimensional quantum field theory. Nucl. Phys. B241, 333 (1984)
Friedan, D., Qiu, Z., Shenker, S.: Details of the non-unitarity proof for highest weight representations of the Virasoro algebra. Commun. Math. Phys.107, 535 (1986); Goddard, P., Kent, A., Olive, D.: Commun. Math. Phys.103, 105 (1986)
Kastor, D., Martinec, E., Qiu, Z.: Phys. Lett.200B, 434 (1988); Bagger, J., Nemeschansky, D., Yankielowicz, S.: Phys. Rev. Lett.60, 389 (1988); Bowcock, P., Goddard, P.: Nucl. Phys. B305, 685 (1988)
Jimbo, M.: Lett. Math. Phys.10, 63 (1985); Drinfel'd, V.G.: Quantum groups, Proc. ICM 798 (1987)
Pasquier, V.: Commun. Math. Phys.118, 355 (1988)
Rehren, K.-H.: Commun. Math. Phys.116, 675 (1988); Rehren, K.-H., Schroer, B.: Einstein Causality and Artin Braids, preprint, FU Berlin, 1988
Gervais, J.-L., Neveu, A.: Nucl. Phys. B238, 125 (1984)
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Felder, G., Fröhlich, J. & Keller, G. On the structure of unitary conformal field theory II: Representation theoretic approach. Commun.Math. Phys. 130, 1–49 (1990). https://doi.org/10.1007/BF02099872
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DOI: https://doi.org/10.1007/BF02099872