This article represents a very substantial review of Kac-Moody and Virasoro algebras and their application to a variety of problems in quantum physics. In these applications highest weight irreducible unitary representations are required and the authors discuss the crucial matter of the values taken in these representations by the central elements in the central extensions of these algebras.
The connection with physics comes about through the assumption of conformal invariance for 2-dimensional quantum field theories. Links with current algebras, \(\sigma\)-models, and statistical systems are all expounded upon. After succinctly summarising the theory of roots, weights and Dynkin diagrams of simple finite-dimensional Lie algebras the authors give an excellent treatment of the same aspects of affine Kac-Moody algebras, both twisted and untwisted. Operator product methods are used to give explicit realisations of Kac-Moody algebras in terms of fermionic fields of the Neveu-Schwarz or Ramond type. The Virasoro algebras are realised via the Sugawara construction in terms of normal ordered products quadratic in the Kac-Moody generators. Vertex operator representations of Kac-Moody algebras are described and compared with quark model representations. Vectorial Fermi fields are used in this context. In the case of so(8), spinorial Fermi fields are also discussed and the triality of so(8) is related to boson-fermion quantum equivalence. The connection with the theory of superstrings is made, wherein, as the authors point out, much remains to be done. As befits such an extensive review a comprehensive list of references is supplied.