Vertex operators in the conformal field theory on \(\mathbb P^1\) and monodromy representations of the braid group
In this note the authors give a rigorous mathematical foundation to the work of \textit{V. G. Knizhnik} and \textit{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)].
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