Geometric, algebraic and topological methods for quantum field theory : proceedings of the 2011 Villa de Leyva Summer School, Villa de Leyva, Colombia, 4-22 July 2011

Based on lectures held at the 7th Villa de Leyva summer school, this book presents an introduction to topics of current interest in the interface of geometry, topology and physics. It is aimed at graduate students in physics or mathematics with interests in geometric, algebraic as well as topological methods and their applications to quantum field theory. This volume contains the written notes corresponding to lectures given by experts in the field. They cover current topics of research in a way that is suitable for graduate students of mathematics or physics interested in the recent developments and interactions between geometry, topology and physics. The book also contains contributions by younger participants, displaying the ample range of topics treated in the school. A key feature of the present volume is the provision of a pedagogical presentation of rather advanced topics, in a way which is suitable for both mathematicians and physicists. Readership: Researchers in geometry and topology, mathematical physics. Introduction; CONTENTS; Part A LECTURES; Spectral Geometry B. Iochum; 1. Motivations; 2. Wodzicki residue and kernel near the diagonal; 2.1. A quick overview on pseudodifferential operators; 2.2. Case of manifolds; 2.3. Singularities of the kernel near the diagonal; 2.4. Wodzicki residue; 3. Dixmier trace; 3.1. Singular values of compact operators; 3.2. Dixmier trace; 4. Dirac operator; 4.1. Definition and main properties; 4.2. Dirac operators and change of metrics; 5. Heat kernel expansion; 5.1. The asymptotics of heat kernel; 5.2. Wodzicki residue and heat expansion