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.

Loop groups are the simplest class of infinite dimensional Lie groups, and have important applications in elementary particle physics. They have recently been studied intensively, and the theory is now well developed, involving ideas from several areas of mathematics - algebra, geometry, analysis, and combinatorics. The mathematics of quantum field theory is an important ingredient. This book gives a complete and self-contained account of what is known about the subject and it is written from a geometrical and analytical point of view, with quantum field theory very much in mind. The mathematics used in connection with loop groups is interesting and important beyond its immediate applications and the authors have tried to make the book accessible to mathematicians in many fields. The hardback edition was published in December 1986.

Eleven of the fourteen invited speakers at a symposium held by the Oxford Mathematical Institute in June 1972 have revised their contributions and submitted them for publication in this volume. The present papers do not necessarily closely correspond with the original talks, as it was the intention of the volume editor to make this book of mathematical rather than historical interest. The contributions will be of value to workers in topology in universities and polytechnics.

Research in string theory has generated a rich interaction with algebraic geometry, with exciting new work that includes the Strominger-Yau-Zaslow conjecture. This monograph builds on lectures at the 2002 Clay School on Geometry and String Theory that sought to bridge the gap between the languages of string theory and algebraic geometry, presenting an updated discussion that includes subsequent developments. The group of distinguished mathematicians and mathematical physicists who produced this monograph worked as a team to create a unique volume. Its overall goal is to explore the physical and mathematical aspects of Dirichlet branes. The narrative is organized around two principal ideas: Kontsevich's Homological Mirror, Symmetry conjecture, and the Strominger-Yau-Zaslow conjecture. The authors explain how Kontsevich's conjecture is equivalent to the identification of two different categories of Dirichlet branes. They also explore the ramifications and current state of the Strominger-Yau-Zaslow conjecture. They relate the ideas to active areas of research that include the McKay correspondence, topological quantum field theory, and stability structures. The authors were not satisfied to tell their story twice, from separate mathematics and physics points of view. Instead, theirs is a unified presentation offered in a way that both mathematicians and physicists can follow, without having all of the foundations of both subjects at their immediate disposal.