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Vertex operators in the conformal field theory on \(\mathbb P^1\) and monodromy representations of the braid group. (English) Zbl 0631.17010

In this note the authors give a rigorous mathematical foundation to the work of V. G. Knizhnik and A. B. Zamolodchikov [Nucl. Phys., B 247, No. 1, 83–103 (1984; Zbl 0661.17020)] on the primary fields for the two-dimensional conformal field theory. One of the results is the existence and uniqueness theorem of primary fields (vertex operators). A multipoint (correlation) function is defined as a vacuum expectation value of the composition of the primary fields. A system of differential equations and a system of algebraic equations are given, which are satisfied by a multipoint function. Using these equations, analytic continuation of a multipoint function is discussed. Finally the authors state the result that the monodromies on multipoint functions coincide with the representations of the Hecke algebra constructed by Wenzl.
Details and proofs will appear in [Conformal field theory and solvable lattice models, Symp., Kyoto/Jap. 1986, Adv. Stud. Pure Math. 16, 297–372 (1988; Zbl 0661.17021)].

MSC:

17B69 Vertex operators; vertex operator algebras and related structures
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
20F36 Braid groups; Artin groups
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References:

[1] BelavinA. A., PolyakovA. N., and ZamolodchikovA. B., ?Infinite Conformal Symmetries in Twodimensional Quantum Field Theory?, Nuclear Phys. B 241, 333-380 (1984). · Zbl 0661.17013
[2] DotsenkoVI. S. and FateevV. A., ?Conformal Algebra and Multipoint Correlation Functions in 2D Statistical Models?, Nuclear Phys. B240, 312-348 (1984).
[3] KacV. G., Infinite Dimensional Lie Algebras, 2nd edn., Cambridge Univ. Press, Cambridge, 1985.
[4] KnizhnikV. G. and ZamolodchikovA. B., ?Current Algebra and Wess-Zumino Models in Two Dimensions?, Nuclear Phys. B247, 83-103 (1984). · Zbl 0661.17020
[5] Wenzl, H., ?Representations of Hecke Algebras and Subfactors?, Thesis, University of Pensylvania, 1985.
[6] ZamolodchikovA. B. and FateevV. A., ?Nonlocal (Parafermion) Currents in Two-dimensional Conformal Quantum Field Theory and Self-dual Critical Points in ? N -Symmetric Statistical Systems?, Sov. Phys. JETP 62(2), 215-225 (1985).
[7] Tsuchiya, A. and Kanie, Y., ?Vertex Operators in Conformal Field Theory on P 1 and Monodromy Representations of Braid Group?, to appear in Conformal Field Theory and Solvable Lattice Models, Advanced Studies in Pure Mathematics, Kinokuniya, Tokyo. · Zbl 0699.17019
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