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Topology and Geometry
in Physics




123
Editors

Eike Bick Frank Daniel Steffen
d-¬ne GmbH DESY Theory Group
Opernplatz 2 Notkestraße 85
60313 Frankfurt 22603 Hamburg
Germany Germany




E. Bick, F.D. Steffen (Eds.), Topology and Geometry in Physics, Lect. Notes Phys. 659 (Springer,
Berlin Heidelberg 2005), DOI 10.1007/b100632




Library of Congress Control Number: 2004116345

ISSN 0075-8450
ISBN 3-540-23125-0 Springer Berlin Heidelberg New York

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Preface




The concepts and methods of topology and geometry are an indispensable part
of theoretical physics today. They have led to a deeper understanding of many
crucial aspects in condensed matter physics, cosmology, gravity, and particle
physics. Moreover, several intriguing connections between only apparently dis-
connected phenomena have been revealed based on these mathematical tools.
Topological and geometrical considerations will continue to play a central role
in theoretical physics. We have high hopes and expect new insights ranging from
an understanding of high-temperature superconductivity up to future progress
in the construction of quantum gravity.
This book can be considered an advanced textbook on modern applications
of topology and geometry in physics. With emphasis on a pedagogical treatment
also of recent developments, it is meant to bring graduate and postgraduate stu-
dents familiar with quantum ¬eld theory (and general relativity) to the frontier
of active research in theoretical physics.
The book consists of ¬ve lectures written by internationally well known ex-
perts with outstanding pedagogical skills. It is based on lectures delivered by
these authors at the autumn school “Topology and Geometry in Physics” held at
the beautiful baroque monastery in Rot an der Rot, Germany, in the year 2001.
This school was organized by the graduate students of the Graduiertenkolleg
“Physical Systems with Many Degrees of Freedom” of the Institute for Theoret-
ical Physics at the University of Heidelberg. As this Graduiertenkolleg supports
graduate students working in various areas of theoretical physics, the topics
were chosen in order to optimize overlap with condensed matter physics, parti-
cle physics, and cosmology. In the introduction we give a brief overview on the
relevance of topology and geometry in physics, describe the outline of the book,
and recommend complementary literature.
We are extremely thankful to Frieder Lenz, Thomas Sch¨cker, Misha Shif-
u
man, Jan-Willem van Holten, and Jean Zinn-Justin for making our autumn
school a very special event, for vivid discussions that helped us to formulate
the introduction, and, of course, for writing the lecture notes for this book.
For the invaluable help in the proofreading of the lecture notes, we would like
to thank Tobias Baier, Kurush Ebrahimi-Fard, Bj¨rn Feuerbacher, J¨rg J¨ckel,
o o a
Filipe Paccetti, Volker Schatz, and Kai Schwenzer.
The organization of the autumn school would not have been possible with-
out our team. We would like to thank Lala Adueva for designing the poster and
the web page, Tobial Baier for proposing the topic, Michael Doran and Volker
VI Preface

Schatz for organizing the transport of the blackboard, J¨rg J¨ckel for ¬nan-
o a
cial management, Annabella Rauscher for recommending the monastery in Rot
an der Rot, and Ste¬en Weinstock for building and maintaining the web page.
Christian Nowak and Kai Schwenzer deserve a special thank for the organiza-
tion of the magni¬cent excursion to Lindau and the boat trip on the Lake of
Constance. The timing in coordination with the weather was remarkable. We
are very thankful for the ¬nancial support from the Graduiertenkolleg “Physical
Systems with Many Degrees of Freedom” and the funds from the Daimler-Benz
Stiftung provided through Dieter Gromes. Finally, we want to thank Franz Weg-
ner, the spokesperson of the Graduiertenkolleg, for help in ¬nancial issues and
his trust in our organization.
We hope that this book has captured some of the spirit of the autumn school
on which it is based.




Heidelberg Eike Bick
July, 2004 Frank Daniel Ste¬en
Contents




Introduction and Overview
E. Bick, F.D. Ste¬en . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1 Topology and Geometry in Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 An Outline of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
3 Complementary Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Topological Concepts in Gauge Theories
F. Lenz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Nielsen“Olesen Vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1 Abelian Higgs Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Topological Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3 Homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1 The Fundamental Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Higher Homotopy Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 Quotient Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.4 Degree of Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.5 Topological Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.6 Transformation Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.7 Defects in Ordered Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4 Yang“Mills Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5 ™t Hooft“Polyakov Monopole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.1 Non-Abelian Higgs Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.2 The Higgs Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.3 Topological Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6 Quantization of Yang“Mills Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
7 Instantons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
7.1 Vacuum Degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
7.2 Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
7.3 Fermions in Topologically Non-trivial Gauge Fields . . . . . . . . . . . . 58
7.4 Instanton Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
7.5 Topological Charge and Link Invariants . . . . . . . . . . . . . . . . . . . . . . . 62
8 Center Symmetry and Con¬nement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
8.1 Gauge Fields at Finite Temperature and Finite Extension . . . . . . . 65
8.2 Residual Gauge Symmetries in QED . . . . . . . . . . . . . . . . . . . . . . . . . 66
8.3 Center Symmetry in SU(2) Yang“Mills Theory . . . . . . . . . . . . . . . . 69
VIII Contents

8.4 Center Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
8.5 The Spectrum of the SU(2) Yang“Mills Theory . . . . . . . . . . . . . . . . 74
9 QCD in Axial Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
9.1 Gauge Fixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
9.2 Perturbation Theory in the Center-Symmetric Phase . . . . . . . . . . . 79
9.3 Polyakov Loops in the Plasma Phase . . . . . . . . . . . . . . . . . . . . . . . . . 83
9.4 Monopoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
9.5 Monopoles and Instantons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
9.6 Elements of Monopole Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
9.7 Monopoles in Diagonalization Gauges . . . . . . . . . . . . . . . . . . . . . . . . 91
10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Aspects of BRST Quantization
J.W. van Holten . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
1 Symmetries and Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
1.1 Dynamical Systems with Constraints . . . . . . . . . . . . . . . . . . . . . . . . 100
1.2 Symmetries and Noether™s Theorems . . . . . . . . . . . . . . . . . . . . . . . . 105
1.3 Canonical Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
1.4 Quantum Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
1.5 The Relativistic Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
1.6 The Electro-magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
1.7 Yang“Mills Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
1.8 The Relativistic String . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
2 Canonical BRST Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
2.1 Grassmann Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
2.2 Classical BRST Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

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