Geometric Methods for Quantum Field Theory : Proceedings of the Summer School Villa de Leyva, Colombia 12-30 July 1999

by Ocampo, Hernan; Reyes, Andres; Paycha, Sylvie

• ISBN13: 9789810243517
• ISBN10: 9810243510
• Format: Hardcover
• Copyright: 2001-06-01
• Publisher: WORLD SCIENTIFIC PUB CO INC

Summary

Both mathematics and mathematical physics have many active areas of research where the interplay between geometry and quantum field theory has proved extremely fruitful. Duality, gauge field theory, geometric quantization, Seiberg -- Witten theory, spectral properties and families of Dirac operators, and the geometry of loop groups offer some striking recent examples of modern topics which stand on the borderline between geometry and analysis on the one hand and quantum field theory on the other, where the physicist's and the mathematician's perspective complement each other, leading to new mathematical and physical concepts and results.This volume introduces the reader to some basic mathematical and physical tools and methods required to follow the recent developments in some active areas of mathematical physics, including duality, gauge field theory, geometric quantization, Seiberg -- Witten theory, spectral properties and families of Dirac operators, and the geometry of loop groups. It comprises seven,self-contained lectures, which should progressively give the reader a precise idea of some of the techniques used in these areas, as well as a few short communications presented by young participants at the school.

Table of Contents

Introduction	v
Lectures
Introduction to differentiable manifolds and symplectic geometry	1	(115)
Tilmann Wurzbacher
Spectral properties of the Dirac operator and geometrical structures	116	(54)
Oussama Hijazi
Quantum theory of fermion systems: Topics between physics and mathematics	170	(68)
Edwin Langmann
Heat equation and spectral geometry. Introduction for beginners	238	(55)
Krzysztof Wojciechowski
Renormalized traces as a geometric tool	293	(68)
Sylvie Paycha
Concepts in gauge theory leading to electric-magnetic duality	361	(60)
Tsou Sheung Tsun
An introduction to Seiberg-Witten theory	421
Hernan Ocampo
Short Communications
Remarks on duality, analytic torsion and gaussian integration in antisymmetric field theories	451
Alexander Cardona
Multiplicative anomaly for the ζ-regularized determinant	467
Catherine Ducourtioux
On cohomogeneity one Riemannian manifolds	483
S. M. B. Kashani
A differentiable calculus on the space of loops and connections	489
Martin Reiris
Quantum Hall conductivity and topological invariants	498
Andres Reyes
Determinant of the Dirac operator over the interval [0, β]	509
Fabian Torres-Ardila