Conformal field theory
Conformal field theory has grown into a subject of mathematical physics in its on right. It was a good idea to write a comprehensive book on the theory which may serve as a general introduction to the subject on the one hand and as a reference manual for the expert on the other. In writing this exceptionally extensive and useful review the authors, however, embedded the central issue (analysis of conformal invariance in two dimensions) in a large, perhaps too large amount of background material: there are introductory chapters on quantum field theory and statistical mechanics as well as finite-size-scaling and the two-dimensional Ising model, followed by mathematical background on simple Lie algebras, their representations, affine Lie algebras, and Verma modules. As for the central part, the reader will find an in-depth discussion of the so-called minimal models, the Coulomb-gas formulation, the BRST cohomology, the role of modular invariance and finite geometries, the Wess-Zumino-Witten models, and fusion rules. This naturally leads to the concept of rational conformal field theory (RCFT), given by fields that are not minimal and whose (possibly infinite) decomposition into primary fields can be reorganized into a finite set of blocks corresponding to an extended symmetry algebra and a central charge \(c>1\). The last chapter is devoted to the coset construction of RCFT's which is expected to provide a complete classification of non-minimal models. Though it covers a lot of material, the text is by no means exhaustive. The authors thus felt that they ought to announce the publication of a second complementary volume covering mainly applications of the present theory, in particular the discussion of two-dimensional quantum gravity.
scientific article; zbMATH DE number 986412