1994/12/29 by W. Boenkost, Wolfram Boenkost, Boenkost, Wolfram
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Nonlinear Waves and Solitons #hep-th
paper · pdf · doi:10.48550/arxiv.hep-th/9412231
118 pages (german) latex
arxiv created 1994/12/29 · arxiv updated 2009/11/30
In this work we describe the mathematical foundations used in the construction of primary fields of minimal models of conformal field theory. The work contains two parts: In the first part we give a description of Verma and Fock modules for the Virasoro algebra and develop their imbedding patterns. This part is a simplification of the work of Feigin and Fuks (we correct a mistake in their patterns in the case III+), Rocha-Charidi and some new ideas which yield a simplification of the original papers. In the second part we define (free) vertex operators as unbounded Hilbert space operators, acting on Fock spaces, which are Virasoro modules. We prove several properties of these operators: under appropriate conditions vertex operators are densely defined, not closable operators. Radially ordered products of vertex operators exists on a dense subset. We prove commutation relations between vertex operators and elements of the Virasoro algebra. Next we define, following the (non rigouros) work of G. Felder, integrated vertex operators and prove that these operators resemble the properties of the not integrated vertex operators. Special integrated vertex operators can be identified with conformal fields and a Virasoro invariant subspace of Fock space can be identified with the physical Hilbert space for the conformal theory.