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2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
2.4 Quantum BRST Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
2.5 BRST-Hodge Decomposition of States . . . . . . . . . . . . . . . . . . . . . . . 138
2.6 BRST Operator Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
2.7 Lie-Algebra Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
3 Action Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
3.1 BRST Invariance from Hamilton™s Principle . . . . . . . . . . . . . . . . . . 146
3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
3.3 Lagrangean BRST Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
3.4 The Master Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
3.5 Path-Integral Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
4 Applications of BRST Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
4.1 BRST Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
4.2 Anomalies and BRST Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . 158
Appendix. Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

Chiral Anomalies and Topology
J. Zinn-Justin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
1 Symmetries, Regularization, Anomalies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
2 Momentum Cut-O¬ Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
Contents IX

2.1 Matter Fields: Propagator Modi¬cation . . . . . . . . . . . . . . . . . . . . . . . 170
2.2 Regulator Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
2.3 Abelian Gauge Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
2.4 Non-Abelian Gauge Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
3 Other Regularization Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
3.1 Dimensional Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
3.2 Lattice Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
3.3 Boson Field Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
3.4 Fermions and the Doubling Problem . . . . . . . . . . . . . . . . . . . . . . . . . 182
4 The Abelian Anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
4.1 Abelian Axial Current and Abelian Vector Gauge Fields . . . . . . . . 184
4.2 Explicit Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
4.3 Two Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
4.4 Non-Abelian Vector Gauge Fields and Abelian Axial Current . . . . 195
4.5 Anomaly and Eigenvalues of the Dirac Operator . . . . . . . . . . . . . . . 196
5 Instantons, Anomalies, and θ-Vacua . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
5.1 The Periodic Cosine Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
5.2 Instantons and Anomaly: CP(N-1) Models . . . . . . . . . . . . . . . . . . . . 201
5.3 Instantons and Anomaly: Non-Abelian Gauge Theories . . . . . . . . . 206
5.4 Fermions in an Instanton Background . . . . . . . . . . . . . . . . . . . . . . . . 210
6 Non-Abelian Anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
6.1 General Axial Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
6.2 Obstruction to Gauge Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
6.3 Wess“Zumino Consistency Conditions . . . . . . . . . . . . . . . . . . . . . . . . 215
7 Lattice Fermions: Ginsparg“Wilson Relation . . . . . . . . . . . . . . . . . . . . . . . 216
7.1 Chiral Symmetry and Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
7.2 Explicit Construction: Overlap Fermions . . . . . . . . . . . . . . . . . . . . . . 221
8 Supersymmetric Quantum Mechanics and Domain Wall Fermions . . . . 222
8.1 Supersymmetric Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 222
8.2 Field Theory in Two Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
8.3 Domain Wall Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
Appendix A. Trace Formula for Periodic Potentials . . . . . . . . . . . . . . . . . . . . . 229
Appendix B. Resolvent of the Hamiltonian in Supersymmetric QM . . . . . . . 231

Supersymmetric Solitons and Topology
M. Shifman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
2 D = 1+1; N = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
2.1 Critical (BPS) Kinks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
2.2 The Kink Mass (Classical) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
2.3 Interpretation of the BPS Equations. Morse Theory . . . . . . . . . . . . 244
2.4 Quantization. Zero Modes: Bosonic and Fermionic . . . . . . . . . . . . . 245
2.5 Cancelation of Nonzero Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
2.6 Anomaly I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
2.7 Anomaly II (Shortening Supermultiplet Down to One State) . . . . 252
3 Domain Walls in (3+1)-Dimensional Theories . . . . . . . . . . . . . . . . . . . . . . 254
X Contents

3.1
Superspace and Super¬elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
3.2
Wess“Zumino Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
3.3
Critical Domain Walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
3.4
Finding the Solution to the BPS Equation . . . . . . . . . . . . . . . . . . . . 261
3.5
Does the BPS Equation Follow from the Second Order Equation
of Motion? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
3.6 Living on a Wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
4 Extended Supersymmetry in Two Dimensions:
The Supersymmetric CP(1) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
4.1 Twisted Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
4.2 BPS Solitons at the Classical Level . . . . . . . . . . . . . . . . . . . . . . . . . . 267
4.3 Quantization of the Bosonic Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . 269
4.4 The Soliton Mass and Holomorphy . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
4.5 Switching On Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
4.6 Combining Bosonic and Fermionic Moduli . . . . . . . . . . . . . . . . . . . . 274
5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
Appendix A. CP(1) Model = O(3) Model (N = 1 Super¬elds N ) . . . . . . . . . 275
Appendix B. Getting Started (Supersymmetry for Beginners) . . . . . . . . . . . . 277
B.1 Promises of Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
B.2 Cosmological Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
B.3 Hierarchy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

Forces from Connes™ Geometry
T. Sch¨cker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
u 285
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
2 Gravity from Riemannian Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
2.1 First Stroke: Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
2.2 Second Stroke: Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
3 Slot Machines and the Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
3.1 Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
3.2 Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
3.3 The Winner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
3.4 Wick Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
4 Connes™ Noncommutative Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
4.1 Motivation: Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
4.2 The Calibrating Example: Riemannian Spin Geometry . . . . . . . . . 305
4.3 Spin Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
5 The Spectral Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
5.1 Repeating Einstein™s Derivation in the Commutative Case . . . . . . 311
5.2 Almost Commutative Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
5.3 The Minimax Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
5.4 A Central Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
6 Connes™ Do-It-Yourself Kit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
6.1 Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
6.2 Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
6.3 The Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
Contents XI

6.4 Beyond the Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
7 Outlook and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
A.1 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
A.2 Group Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342
A.3 Semi-Direct Product and Poincar´ Group . . . . . . . . . . . . . . . . . . . . .
e 344
A.4 Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
List of Contributors




Jan-Willem van Holten
National Institute for Nuclear and High-Energy Physics
(NIKHEF)
P.O. Box 41882
1009 DB Amsterdam, the Netherlands
and
Department of Physics and Astronomy
Faculty of Science
Vrije Universiteit Amsterdam
t32@nikhef.nl
Frieder Lenz
Institute for Theoretical Physics III
University of Erlangen-N¨rnberg
u
Staudstrasse 7
91058 Erlangen, Germany
flenz@theorie3.physik.uni-erlangen.de
Thomas Sch¨ cker
u
Centre de Physique Th´orique
e
CNRS - Luminy, Case 907
13288 Marseille Cedex 9, France
Thomas.Schucker@cpt.univ-mrs.fr
Mikhail Shifman
William I. Fine Theoretical Physics Institute
University of Minnesota
116 Church Street SE
Minneapolis MN 55455, USA
shifman@umn.edu
Jean Zinn-Justin
Dapnia
CEA/Saclay
91191 Gif-sur-Yvette Cedex, France
jean.zinn-justin@cea.fr
Introduction and Overview

E. Bick1 and F.D. Ste¬en2
1
d-¬ne GmbH, Opernplatz 2, 60313 Frankfurt, Germany
2
DESY Theory Group, Notkestr. 85, 22603 Hamburg, Germany



1 Topology and Geometry in Physics
The ¬rst part of the 20th century saw the most revolutionary breakthroughs in
the history of theoretical physics, the birth of general relativity and quantum
¬eld theory. The seemingly nearly completed description of our world by means
of classical ¬eld theories in a simple Euclidean geometrical setting experienced
major modi¬cations: Euclidean geometry was abandoned in favor of Rieman-
nian geometry, and the classical ¬eld theories had to be quantized. These ideas
gave rise to today™s theory of gravitation and the standard model of elemen-
tary particles, which describe nature better than anything physicists ever had at
hand. The dramatically large number of successful predictions of both theories
is accompanied by an equally dramatically large number of problems.
The standard model of elementary particles is described in the framework
of quantum ¬eld theory. To construct a quantum ¬eld theory, we ¬rst have to
quantize some classical ¬eld theory. Since calculations in the quantized theory are
plagued by divergencies, we have to impose a regularization scheme and prove
renormalizability before calculating the physical properties of the theory. Not
even one of these steps may be carried out without care, and, of course, they
are not at all independent. Furthermore, it is far from clear how to reconcile
general relativity with the standard model of elementary particles. This task
is extremely hard to attack since both theories are formulated in a completely
di¬erent mathematical language.
Since the 1970™s, a lot of progress has been made in clearing up these di¬cul-
ties. Interestingly, many of the key ingredients of these contributions are related
to topological structures so that nowadays topology is an indispensable part of
theoretical physics.
Consider, for example, the quantization of a gauge ¬eld theory. To quantize
such a theory one chooses some particular gauge to get rid of redundant degrees
of freedom. Gauge invariance as a symmetry property is lost during this process.
This is devastating for the proof of renormalizability since gauge invariance is
needed to constrain the terms appearing in the renormalized theory. BRST quan-
tization solves this problem using concepts transferred from algebraic geometry.
More generally, the BRST formalism provides an elegant framework for dealing
with constrained systems, for example, in general relativity or string theories.
Once we have quantized the theory, we may ask for properties of the classical
theory, especially symmetries, which are inherited by the quantum ¬eld theory.
Somewhat surprisingly, one ¬nds obstructions to the construction of quantized

E. Bick and F.D. Ste¬en, Introduction and Overview, Lect. Notes Phys. 659, 1“5 (2005)
http://www.springerlink.com/ c Springer-Verlag Berlin Heidelberg 2005
2 E. Bick and F.D. Ste¬en

gauge theories when gauge ¬elds couple di¬erently to the two fermion chiral
components, the so-called chiral anomalies. This puzzle is connected to the dif-
¬culties in regularizing such chiral gauge theories without breaking chiral sym-
metry. Physical theories are required to be anomaly-free with respect to local
symmetries. This is of fundamental signi¬cance as it constrains the couplings
and the particle content of the standard model, whose electroweak sector is a
chiral gauge theory.
Until recently, because exact chiral symmetry could not be implemented on
the lattice, the discussion of anomalies was only perturbative, and one could
have feared problems with anomaly cancelations beyond perturbation theory.
Furthermore, this di¬culty prevented a numerical study of relevant quantum
¬eld theories. In recent years new lattice regularization schemes have been dis-
covered (domain wall, overlap, and perfect action fermions or, more generally,
Ginsparg“Wilson fermions) that are compatible with a generalized form of chiral
symmetry. They seem to solve both problems. Moreover, these lattice construc-
tions provide new insights into the topological properties of anomalies.
The questions of quantizing and regularizing settled, we want to calculate the
physical properties of the quantum ¬eld theory. The spectacular success of the
standard model is mainly founded on perturbative calculations. However, as we
know today, the spectrum of e¬ects in the standard model is much richer than
perturbation theory would let us suspect. Instantons, monopoles, and solitons
are examples of topological objects in quantum ¬eld theories that cannot be un-
derstood by means of perturbation theory. The implications of this subject are
far reaching and go beyond the standard model: From new aspects of the con-
¬nement problem to the understanding of superconductors, from the motivation
for cosmic in¬‚ation to intriguing phenomena in supersymmetric models.
Accompanying the progress in quantum ¬eld theory, attempts have been
made to merge the standard model and general relativity. In the setting of non-
commutative geometry, it is possible to formulate the standard model in geo-
metrical terms. This allows us to discuss both the standard model and general
relativity in the same mathematical language, a necessary prerequisite to recon-
cile them.


2 An Outline of the Book

This book consists of ¬ve separate lectures, which are to a large extend self-
contained. Of course, there are cross relations, which are taken into account by
the outline.
In the ¬rst lecture, “Topological Concepts in Gauge Theories,” Frieder Lenz
presents an introduction to topological methods in studies of gauge theories.
He discusses the three paradigms of topological objects: the Nielsen“Olesen vor-
tex of the abelian Higgs model, the ™t Hooft“Polyakov monopole of the non-
abelian Higgs model, and the instanton of Yang“Mills theory. The presentation
emphasizes the common formal properties of these objects and their relevance

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