Loop groups, the simplest class of infinite dimensional Lie groups, have recently been the subject of intense study. This book gives a complete and self-contained account of what is known about them from a geometrical and analytical point of view, drawing together the many branches of mathematics from which current theory developed--algebra, geometry, analysis, combinatorics, and the mathematics of quantum field theory. The authors discuss loop groups' applications to simple particle physics and explain how the mathematics used in connection with loop groups is itself interesting and valuable, thereby making this work accessible to mathematicians in many fields.

Introduction; PART 1 - Finite dimensional lie groups; Groups of smooth maps; Central extensions; The root system: KAC-Moody algebras; Loop groups as groups of operators in Hilbert space; The Grassmannian of Hilbert space and the determinant line bundle; The fundamental homogeneous space. PART 2 - Representation theory; The fundamental representation; The Borel-Weil theory; The spin representation; 'Blips' or 'vertex operators'; The KAC character formula and the Bernstein-Gelfand-Gelfand resolution; References; Index of notation; Index.