Differential geometry, in the classical sense, is developed through the theory of smooth manifolds. Modern differential geometry from the author’s perspective is used in this work to describe physical theories of a geometric character without using any notion of calculus (smoothness). Instead, an axiomatic treatment of differential geometry is presented via sheaf theory (geometry) and sheaf cohomology (analysis). Using vector sheaves, in place of bundles, based on arbitrary topological spaces, this unique approach in general furthers new perspectives and calculations that generate unexpected potential applications.
<EM>Modern Differential Geometry in Gauge Theories is a two-volume research monograph that systematically applies a sheaf-theoretic approach to such physical theories as gauge theory. Beginning with Volume 1, the focus is on Maxwell fields. All the basic concepts of this mathematical approach are formulated and used thereafter to describe elementary particles, electromagnetism, and geometric prequantization. Maxwell fields are fully examined and classified in the language of sheaf theory and sheaf cohomology. Continuing in Volume 2, this sheaf-theoretic approach is applied to Yang–Mills fields in general.
The text contains a wealth of detailed and rigorous computations and will appeal to mathematicians and physicists, along with advanced undergraduate and graduate students, interested in applications of differential geometry to physical theories such as general relativity, elementary particle physics and quantum gravity.
Consulting Editor George A. Anastassiou Department of Mathematical Sciences University of Memphis Anastasios Mallios Modern Differential Geometry in Gauge Theories Yang–Mills Fields, Volume II Birkhäuser Boston • Basel • Berlin Anastasios Mallios Department of Mathematics University of Athens Panepistimioupolis GR-157 84, Athens, Greece
[email protected] ISBN 978-0-8176-4379-9 e-ISBN 978-0-8176-4634-9 DOI 10.1007/978-0-8176-4634-9 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2009931685 Mathematics Subject Classification (2000): 53C05, 53C07, 53C80, 18F20, 53D50, 53Z05, 55N30, 19M05, 58A40, 58D27, 58D30, 58K99, 58Z05, 58E15, 81Q70, 81P99, 81T13, 83C45, 83C47, 16D10, 16D40, 16E99, 58C99, 55R05. © Birkhäuser Boston, a part of Springer Science+Business Media, LLC 2010 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhäuser Boston, c/o Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Birkhäuser Boston is part of Springer Science+Business Media (www.birkhauser.com) Contents General Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Preface to Volume II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv Contents of Volume I . . . . . . .