Now both theories have gamed considerable interest in their respective fields, mathematics and physics. Despite the fact that these theories have no common motivations, goals or problems, their formal similarity goes remarkably far. In this paper we discuss primarily the mathematical theory. For a review of the physical theory see the paper of J. Schwarz in this volume [27]
Then in [9, 28] the "vertex construction" was found for the whole class of affine Lie algebras and the similarity became a precise correspondence.
