The Beacon of Kac-moody Symmetry for Physics

Although the importance of developing a mathematics which transcends practical use was already understood by the Greeks over 2000 years ago, it is heartening even today when mathematical ideas created for their abstract interest are found to be useful in formulating descriptions of nature. Historically, the idea of symmetry has its scientific origin in the Greeks' discovery of the five regular solids, which are remarkably symmetrical. In the 19th century, this property was codified in the mathematical concept of a group invented by Galois and then that of a continuous group by Sophus Lie. In 1967, Victor Kac (MIT) then working in Moscow [K] and Bob Moody (Alberta) [M] independently enlarged the paradigm of classical Lie algebras, resulting in new algebras which are infinite-dimensional. The representation theory of a subclass of the algebras, the affine Kac-Moody algebras, has developed into a mature mathematics. By the 1980's, these algebras had been taken up by physicists working in the areas of elementary particle theory, gravity, and two-dimensional phase transitions as an obvious framework from which to consider descriptions of non-perturbative solutions of gauge theory, vertex emission operators in string theory on compactified space, integrability in two-dimensional quantum field theory, and conformal field theory. Recently Kac-Moody algebras have been shown to serve as duality symmetries of non-perturbative strings appearing to relate all superstrings to a single theory. The infinite-dimensional Lie algebras and groups have been suggested as candidates for a unified symmetry of superstring theory. In addition to this wide application to physical theories, the Kac-Moody algebras are relevant to number theory and modular forms. They occur in the relation between the sporadic simple Monster group and the symmetries of codes, lattices, and conformal field theories [Gr,CN]. Much in the same way that certain finite groups were understood to be the symmetry groups of the regular solids, the Monster and certain affine Lie algebras are seen to be automorphisms of conformal field theories [FLM,DGM]. Kac's generalization of Weyl's character formula for the affine algebras leads to a deeper understanding of Mac-donald combinatoric identities relating infinite products and sums [Mac,K1]. Topological properties of the groups corresponding to the affine algebras have been analyzed [Ge,PS], as well as varieties related to the the singularity theory of Kac-Moody algebras [K2,Na,Sl]. Several reviews emphasize the physical applications of the algebras [D,GO,J].