Geometric Analysis and Applications to Quantum Field Theory

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