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This book is detacided ot the fond memory of my .rehtom
Topology, Riemann,
dna Physics
Second Edition
Michael yksrytsanoM
With a Foreword by
Freeman .J Dyson
yb
Translated
Roger Cooke
James King
Victoria King
Reprint of the 1999 Second Edition
resui~hkriB
Boston (cid:12)9 Basel ~ Berlin
Michael Monastyrsky Roger Cooke (Translator)
Department of Theoretical Physics Department of Mathematics
Institute of Theoretical and and Statistics
ExperimentalP hysics University of Vermont
Moscow 117529 Burlington, VT 04505
Russia U.S.A.
James King (Translator)
Victoria King (Translator)
7326 55th Avenue N.E.
Seattle, WA 98115 U.S.A.
Originally published as a monograph
ISBN-13:978-0-8176-4778-0 e-ISBN-13:978-0-8176-4779-7
DOI: 10.1007/978-0-8176-4779-7
Library of Congress Control Number: 2007940672
Mathematics Subject Classification (2000): 01Axx, 12-02, 53-02, 12-03, 53-03, 55-02, 55-03, 70-02,
70-03
@2008 resui~hkriB Boston
All rights reserved. This work may notb e translated or copied in whole or in part without the writ-
ten permission of the publisher Boston, (Birkhiiuser c/o Springer Science+Business Media LLC, 332
Spring Street, New NY York, 10013, USA), except for brief excerpts in connection with reviews or
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Printed on acid-free paper.
987654321
www. birkhauser, com
Michael Monastyrsky
,nnameiR ,ygolopoT
dna Physics
Second Edition
With a Foreword by Freeman J. Dyson
detalsnarT by
Roger Cooke
James King
Victoria King
Birkh~iuser
Boston (cid:12)9 Basel (cid:12)9 Berlin
Michael Monastyrsky
)rotalsnarT(
Department of Theoretical Physics Roger Cooke
Insitute for Theoretical and Department of Mathematics & Statistics
Experimental Physics University of Vermont
Moscow 117529, Russia Burlington, VT 04505
)rotalsnarT(
James King
)rotalsnarT(
Victoria King
7326 55th Avenue N.E.
Seattle, WA 98115
Library of Data Cataloging-in-Publication Congress
Monastyrski~, Mikhail Irich.
[Bemkhard Riman. English]
Riemann, topology, and physics / Michael Monastyrsky ; with a
foreword by Freeman J. Dyson. -- 2nd ed. / translated by Roger
Cooke.
p. cm.
Includes bibliographical references and index.
ISBN 0-8176-3789-3 (alk. paper). -- ISBN 3-7643-3789-3 (alk.
paper)
.1 Riemann, Bernhard, 1826--1866. 2. Topology. 3. Mathematical
physics. 4. Mathematicians---Germany--Biography. I. Title.
QA29.R425M6613 1998
530.15---dc21 98-7034
CIP
AMS Subject Classifications: 01Axx 112-02112-03, 53-02, 53-03, 55-02, 55-03170-02170-03
Printed on acid-free paper.
resui~hkriB ~ |
(cid:14)9 Birkh~iuser Boston, 2 '~ edition.
(cid:14)9 Birkhtiuser Boston, 1 ts edition.
Part I of the ~ 1 edition previously appeared as Bernhard Riemann, (cid:14)9 lzdatel'stvo "Znanije,"
Moscow, 1979.
All rights reserved. This work may not be translated or copied in whole or in part without the written
permission of the publisher (Birkh~iuser Boston, c/o Springer-Verlag New York, Inc., 175 Fifth Avenue,
New York, NY 10010, 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 of general descriptive names, trade names, trademarks, etc., in this publication, even if the former
are not especially identifieids, not to be taken as a sign that such names, as understood by the Trade Marks
and Merchandise Marks Act, may accordingly be used freely by anyone.
ISBN 0-8176-3789-3
ISBN 3-7643-3789-3
Reformattedf rom translator's disk by TEXniques, Inc., Cambridge, MA.
Printed and bound by Quinn-Woodbine, Woodbine, NJ.
Printed in the United States of America.
987654321
Contents
Preface to the Second Edition ............................................... vii
Foreword to the First Edition ................................................. ix
From the Introduction to the First Edition ..................................... xi
Acknowledgements ........................................................ xv
Bernhard Riemann .................................................... 1
Chapter 1 Beginnings ................................................. 3
Chapter 2 Doctoral Dissertation ....................................... 11
Chapter 3 Privatdozent at G6ttingen ................................... 23
Chapter 4 Riemann and Dirichlet ...................................... 43
Chapter 5 Full Professor in G6ttingen .................................. 63
Chapter 6 Last Years ................................................. 75
Chapter 7 Posthumous Fate ........................................... 77
Bibliography to Part I ................................................... 83
II Topological Themes in Contemporary Physics ......................... 85
Chapter 8 Introduction ............................................... 89
Chapter 9 Topological Structures ...................................... 95
Chapter 10 The Connectivity of a Manifold and
Quantization of Magnetic Flux .............................. 107
Chapter 11 Systems with Spontaneous Symmetry Breaking .............. 109
Chapter 12 Topology and Liquid Crystals .............................. 113
Chapter 13 Theory of Gauge Fields .................................... 127
Chapter 14 Topological Particles ...................................... 145
14.1 Instantons, or Pseudo-Particles ........................ 149
vi Contents
Chapter 15 Soliton Particles ........................................... 157
Chapter 16 Knots, Links, and Physics .................................. 167
16.1 Von Neumann Algebras ............................... 168
16.1.1 The Dimension of a Factor .......................... 169
16.2 The Theory of Braids ................................. 171
16.2.1 The Group of Braids ................................. 172
16.2.2 The Skein Relations ................................ 174
16.3 Condensed Matter Theory ............................. 176
16.3.1 Example 1 ......................................... 177
16.3.2 Example 2 ......................................... 179
16.3.3 Magnetohydrodynamics ............................. 179
Chapter 17 What Next? ............................................... 183
17.1 The Quantum Hall Effect .............................. 184
17.2 Quasicrystals ......................................... 185
17.3 Membranes .......................................... 188
Chapter 18 A Brief Historical Survey .................................. 191
Bibliography to Part II ................................................. 195
Index ................................................................. 199
Preface to the Second Edition
Ten years have passed since the publication of the first English edition. "Ten years is
a long time, even when you are not in jail." If we recall that the article on Riemann
appeared in adorir P (Nature) in 1976, this Russian vaudeville joke is more than ap-
propriate. Only the author's youth can account for the insane enterprise of presenting
the scientific achievements and the biography of Riemann in 60 pages and the con-
nection between physics and topology in the same space. Judging from the fact that
the book sold out and got favorable reviews, there is a demand for publications of
this type. It seems to me that popularizations aimed not at a narrow specialist but at
a broader reader are especially needed nowadays. Specialists obtain information in
their fields almost instantly thanks to the Internet, but to find out what is happening
in contiguous fields, what problems and results are of great interest here, is not at all
easy.
Returning to my own book, I note with a certain pride that at least I seem to
have evaluated accurately the trends in the development of theoretical physics. It is
the combination of physics with topology and algebraic geometry that has led to the
brightest achievements of mathematical physics in the last decade. It suffices to note
the remarkable results of S. Donaldson on 4-dimensional smooth manifolds obtained
in the early 1980's and closely connected with the theory of gauge fields (instantons)
and the very recent papers of E. Witten and N. Seiberg, which are also based on
physical ideas and are shedding new light on Donaldson's theory.
A second indisputable example is the development of knot theory after the re-
markable 1984 paper of .V Jones. Here also the main progress is closely connected
with the intertwining of ideas and methods from field theory, statistical physics, and
topology. I have tried to include some of the new advances in the second edition,
keeping in mind the popular level on which this book is written.
Now a few words about the first part, in which the scientific biography of Riemann
is discussed. This part has undergone only minor changes. I have corrected certain
historical and mathematical inaccuracies and a large number of misprints, and added
a few new facts about the development of Riemann's ideas. Indeed the last decade
has confirmed the amazing modernity and fecundity of Riemann's ideas and results.
The most sensational result of recent years has been the proof of Fermat's last
theorem. The proof of A. Wiles and R. Taylor uses the full power of modern algebraic
geometry, whose foundation played a definitive role for Riemann's ideas and methods.
