Softcover ISBN:  9780821821305 
Product Code:  MMONO/210 
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Electronic ISBN:  9781470446352 
Product Code:  MMONO/210.E 
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Book DetailsTranslations of Mathematical MonographsIwanami Series in Modern MathematicsVolume: 210; 2002; 172 ppMSC: Primary 54; 14; Secondary 46; 20;
One of the most remarkable interactions between geometry and physics since 1980 has been an application of quantum field theory to topology and differential geometry. An essential difficulty in quantum field theory comes from infinitedimensional freedom of a system. Techniques dealing with such infinitedimensional objects developed in the framework of quantum field theory have been influential in geometry as well. This book focuses on the relationship between twodimensional quantum field theory and threedimensional topology which has been studied intensively since the discovery of the Jones polynomial in the middle of the 1980s and Witten's invariant for 3manifolds which was derived from ChernSimons gauge theory. This book gives an accessible treatment for a rigorous construction of topological invariants originally defined as partition functions of fields on manifolds.
The book is organized as follows: The Introduction starts from classical mechanics and explains basic background materials in quantum field theory and geometry. Chapter 1 presents conformal field theory based on the geometry of loop groups. Chapter 2 deals with the holonomy of conformal field theory. Chapter 3 treats ChernSimons perturbation theory. The final chapter discusses topological invariants for 3manifolds derived from ChernSimons perturbation theory.ReadershipGraduate students and research mathematicians interested in topology and algebraic geometry.

Table of Contents

Chapters

Introduction

Geometric aspects of conformal field theory

JonesWitten theory

ChernSimons perturbation theory

Further developments and prospects


Reviews

All together, this book should be regarded as a highly valuable and welcome contribution … The author has succeeded in presenting a treatise on this ultramodern and mathematically utmost challenging subject … particularly captivating by its conciseness, clarity, systematic representation, mathematical rigour, and endeavour to unify various related approaches … an excellent source of knowledge and inspiration for both physicists and mathematicians, researchers and seasoned graduate students … those readers will undoubtedly acknowledge the mastery of this book.
Zentralblatt MATH


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One of the most remarkable interactions between geometry and physics since 1980 has been an application of quantum field theory to topology and differential geometry. An essential difficulty in quantum field theory comes from infinitedimensional freedom of a system. Techniques dealing with such infinitedimensional objects developed in the framework of quantum field theory have been influential in geometry as well. This book focuses on the relationship between twodimensional quantum field theory and threedimensional topology which has been studied intensively since the discovery of the Jones polynomial in the middle of the 1980s and Witten's invariant for 3manifolds which was derived from ChernSimons gauge theory. This book gives an accessible treatment for a rigorous construction of topological invariants originally defined as partition functions of fields on manifolds.
The book is organized as follows: The Introduction starts from classical mechanics and explains basic background materials in quantum field theory and geometry. Chapter 1 presents conformal field theory based on the geometry of loop groups. Chapter 2 deals with the holonomy of conformal field theory. Chapter 3 treats ChernSimons perturbation theory. The final chapter discusses topological invariants for 3manifolds derived from ChernSimons perturbation theory.
Graduate students and research mathematicians interested in topology and algebraic geometry.

Chapters

Introduction

Geometric aspects of conformal field theory

JonesWitten theory

ChernSimons perturbation theory

Further developments and prospects

All together, this book should be regarded as a highly valuable and welcome contribution … The author has succeeded in presenting a treatise on this ultramodern and mathematically utmost challenging subject … particularly captivating by its conciseness, clarity, systematic representation, mathematical rigour, and endeavour to unify various related approaches … an excellent source of knowledge and inspiration for both physicists and mathematicians, researchers and seasoned graduate students … those readers will undoubtedly acknowledge the mastery of this book.
Zentralblatt MATH