Vertex operators in conformal field theory on \({\mathbb{P}}^ 1\) and monodromy representations of braid group. (English) Zbl 0661.17021

Conformal field theory and solvable lattice models, Symp., Kyoto/Jap. 1986, Adv. Stud. Pure Math. 16, 297-372 (1988).
[For the entire collection see Zbl 0646.00016.]
The paper is a very clear exposition of the theory of vertex operators in two-dimensional conformal field theory, and the relation with monodromy representations of the braid group. The ideas presented here originate from a paper by V. G. Knizhnik and A. B. Zamolodchikov [Nucl. Phys. B 247, 83-103 (1984; Zbl 0661.17020)], but the present paper goes much deeper in the mathematical formulation and the foundations of the theory. For people who wish not to read such a long paper, there is a 10- page introduction listing the notation and all the basic theorems proved here.
The paper gives a summary of representation theory of the affine sl(2,\({\mathbb{C}})\) Lie algebra, denoted by \(A_ 1^{(1)}\), including the natural action of the operators of the Virasoro algebra. Then, the existence and uniqueness of vertex operators of spin j is shown: vertex operators are defined as operators satisfying the gauge condition and the equations of motion. It is shown that a triplet \(\left( \begin{matrix} j\\ j_ 2\quad j_ 1\end{matrix} \right)\), satisfying the \(\ell\)-constrained Clebsch-Gordan condition, determines uniquely a vertex operator. Next, N- point functions are defined as compositions of vertex operators. Differential equations (called the fundamental equations) satisfied by these N-point functions, which have only regular singularities, give rise to properties of the vertex operators. First, convergence of compositions of vertex operators is proven. The commutation relation of vertex operators is equivalently rephrased in terms of the connection matrix of the fundamental equations, and is explicitly calculated in the case where one of the j-values is equal to \(1/2\). Then, the monodromies of the fundamental equations give rise to representations of the braid group \(B_ N\). The monodromy representation is explicitly determined in a special case (where implicitly \(j=\) vertex operators are involved). This representation is in fact an irreducible representation of the Hecke algebra \(H_ N(q)\) of type \(A_{N-1}\), with \(q=\exp (2\pi \sqrt{- 1}/(\ell +2))\). It is remarkable that the representation obtained here is with q a root of unity, since the Hecke algebra \(H_ N(q)\) is not semi- simple in that case.
Reviewer: J.Van der Jeugt


17B65 Infinite-dimensional Lie (super)algebras
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
20F36 Braid groups; Artin groups