Peter Bouwknegt - Böcker

This volume includes articles on the interface of geometry and mathematical physics that are based on lectures delivered at the University of Adelaide, with an audience of primarily graduate students. The aim is to provide surveys of progress, without assuming too much prerequisite knowledge, so that researchers and graduate students in geometry and mathematical physics will benefit.The contributors cover a number of areas in mathematical physics: Chapter 1 offers a self-contained derivation of the partition function of Chern-Simons gauge theory in the semiclassical approximation; Chapter 2 considers the algebraic and geometric aspects of the Knizhnik-Zamolodchikov equations in conformal field theory, including their relation to the braid group, quantum groups and infinite dimensional Lie algebras; Chapter 3 surveys the application of the represenation theory of loop groups to simple models in quantum field theory and to certain integrable systems; Chapter 4 examines the variational methods in Hermitian geometry from the viewpoint of the critical points of action functionals together with physical backgrounds; Chapter 5 is a review of monopoles in non-Abelian gauge theories and the various approaches to understanding them; Chapter 6 covers much of the exciting recent developments in quantum cohomology, including relative Gromov-Witten invariant, birational geometry, naturality and mirror symmetry; Chapter 7 explains the physics origin of the Seiberg-Witten equations in four-manifold theory and a number of important concepts in quantum-field theory, such as vac

In the last decade there has been an extraordinary confluence of ideas in mathematics and theoretical physics brought about by pioneering discoveries in geometry and analysis. The various chapters in this volume, treating the interface of geometric analysis and mathematical physics, represent current research interests. No suitable succinct account of the material is available elsewhere. Key topics include: * A self-contained derivation of the partition function of Chern- Simons gauge theory in the semiclassical approximation (D.H. Adams) * Algebraic and geometric aspects of the Knizhnik-Zamolodchikov equations in conformal field theory (P. Bouwknegt) * Application of the representation theory of loop groups to simple models in quantum field theory and to certain integrable systems (A.L. Carey and E. Langmann) * A study of variational methods in Hermitian geometry from the viewpoint of the critical points of action functionals together with physical backgrounds (A. Harris) * A review of monopoles in nonabelian gauge theories (M.K. Murray) * Exciting developments in quantum cohomology (Y. Ruan) * The physics origin of Seiberg-Witten equations in 4-manifold theory (S. Wu) Graduate students, mathematicians and mathematical physicists in the above-mentioned areas will benefit from the user-friendly introductory style of each chapter as well as the comprehensive bibliographies provided for each topic. Prerequisite knowledge is minimal since sufficient background material motivates each chapter.

The study of W algebras began in 1985 in the context of two-dimensional conf- mal field theories, the aim being to explore higher-spin extensions of the Virasoro algebra. Given the simultaneous growth in the understanding of two-dimensional metric gravity inspired by analyses of string models, it was inevitable that these algebras would be applied to give analogues of putative higher-spin gravity t- ories. This book is an exposition of the past few years of our work on such an application for the algebra: in particular, the BRST quantization of the n- critical 4D string. We calculate the physical spectrum as a problem in BRST cohomology. The corresponding operator cohomology forms a BV algebra, for which we provide a geometrical model. The algebra has one further generator, of spin three, in addition to the (spin two) energy-momentum tensor which generates the Virasoro algebra. C- trary to the Virasoro algebra, it is an algebra defined by nonlinear relations. In deriving our understanding of the resulting gravity theories we have had to - velop a number of results on the representation theory of W algebras, to replace the standard techniques that were so successful in treating linear algebras.

W-symmetry is an extension of conformal symmetry in two dimensions. Since its introduction in 1985, W-symmetry has become one of the central notions in the study of two-dimensional conformal field theory. The mathematical structures that underlie W-symmetry are so-called W-algebras, which are higher-spin extensions of the Virasoro algebra. This book contains a collection of papers on W-symmetry, covering the period from 1985 through 1993. Its main focus is the construction of W-algebras and their representation theory. A recurrent theme is the intimate connection between W-algebras and affine Lie algebras. Some of the applications, in particular W-gravity, are also covered.The significance of this reprint volume is that there are no textbooks entirely devoted to the subject.

The following topics were covered: the study of renormalization group flows between field theories using the methods of quantum integrability, S-matrix theory and the thermodynamic Bethe Ansatz; impurity problems approached both from the point of view of conformal field theory and quantum integrability. This includes the Kondo effect and quantum wires; solvable models with 1/r² interactions (Haldane-Shastri models). Yangian symmetries in 1/r² models and in conformal field theories; correlation functions in integrable 1+1 field theories; integrability in three dimensions; conformal invariance and the quantum hall effect; supersymmetry in statistical mechanics; and relations to two-dimensional Yang-Mills and QCD.