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Operator algebras and conformal field theory. III: Fusion of positive energy representations of \(LSU(N)\) using bounded operators. (English) Zbl 0944.46059

This mammoth work is one of a series of papers devoted to the study of conformal field theory from the point of view of operator algebras [see also Proc. ICM ’94, Zürich, 966-979 (1995; Zbl 0854.46055)]. It stems from the algebraic approach to quantum field theory of Doplicher, Haag and Roberts. Taking the conformal group as symmetry group leads to conformal field theory (CFT). The case of \(1+1\) dimensional space-time is especially interesting, since the conformal group is then infinite-dimensional. In this case, CFT’s are described by a net of algebras \(A(I)\), indexed by a set of proper intervals \(I\subset S^1\), and satisfying the corresponding axioms such as isotony, locality, conformal symmetry and conformal covariance. The whole system is described by the \(C^*\)-inductive limit \(A\) of the net. Local quantum fields consist of operator-valued functions \(\phi(z, \overline{z})\) on the Minkowski space, with the corresponding operators acting on a Hilbert space \(H\). The fields are often identified, by virtue of Reeh-Schlieder theorem, with vectors from \(H\), and also with the representations generated by the corresponding vector states. The vectors leading to a highest weight representation, corresponding to the lowest energy, give rise to primary fields.
The author presents a fermionic construction of primary fields and shows that all vector primary fields \(\Phi\) can be obtained by ‘compressing’ the smeared fermion field \(a(f)\), \(f\in L^2(S^1, {\mathbb C}^N)\). This yields an automatic \(L^2\) bound \(\|\Phi(f)\|\leq \|f\|_2\). The fermionic construction leads also to examples of nontrivial algebraic quantum field theories in 2 dimensions with finitely many sectors and non-integer statistical dimension. The highly nontrivial problem of CFT is how to combine (‘fuse’) representations in terms of products of primary fields. In the paper, the positive energy representations of the loop groups \(L\text{SU}(N)= C^\infty(S^1, \text{SU}(N))\) are considered. These are obtained by composing the vacuum representation \(\pi_0\) with specific endomorphisms \(\rho\) of the algebra. The fusion of superselection sectors, i.e. equivalence classes \([\pi_0\circ\rho_i]\), \(i=1,2\), should yield \([\pi_0\circ\rho_1\circ \rho_2]\). The positive energy representations are related to bimodules over von Neumann algebras and their fusion to Connes’ fusion, i.e. tensor product operation on such bimodules.
The detailed analysis of primary fields is complemented by a thorough study of the corresponding differential equations, called transport formulas. The outcome is the explicit computation of the Connes fusion for positive energy representation. The paper clarifies the interplay of the Doplicher-Haag-Roberts algebraic quantum field theory, Connes’ theory of bimodules and Jones’ theory of subfactors, showing their connections with the theory of primary fields of CFT.

MSC:

46L60 Applications of selfadjoint operator algebras to physics
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
81T05 Axiomatic quantum field theory; operator algebras
81T10 Model quantum field theories

Citations:

Zbl 0854.46055
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