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in physics. For example, our understanding of superconductivity based on the
Introduction and Overview 3

abelian Higgs model, or Ginzburg“Landau model, is described. A compact re-
view of Yang“Mills theory and the Faddeev“Popov quantization procedure of
gauge theories is given, which addresses also the topological obstructions that
arise when global gauge conditions are implemented. Our understanding of con-
¬nement, the key puzzle in quantum chromodynamics, is discussed in light of
topological insights. This lecture also contains an introduction to the concept of
homotopy with many illustrating examples and applications from various areas
of physics.
The quantization of Yang“Mills theory is revisited as a speci¬c example in the
lecture “Aspects of BRST Quantization” by Jan-Willem van Holten. His lecture
presents an elegant and powerful framework for dealing with quite general classes
of constrained systems using ideas borrowed from algebraic geometry. In a very
systematic way, the general formulation is always described ¬rst, which is then
illustrated explicitly for the relativistic particle, the classical electro-magnetic
¬eld, Yang“Mills theory, and the relativistic bosonic string. Beyond the pertur-
bative quantization of gauge theories, the lecture describes the construction of
BRST-¬eld theories and the derivation of the Wess“Zumino consistency condi-
tion relevant for the study of anomalies in chiral gauge theories.
The study of anomalies in gauge theories with chiral fermions is a key to most
fascinating topological aspects of quantum ¬eld theory. Jean Zinn-Justin de-
scribes these aspects in his lecture “Chiral Anomalies and Topology.” He reviews
various perturbative and non-perturbative regularization schemes emphasizing
possible anomalies in the presence of both gauge ¬elds and chiral fermions. In
simple examples the form of the anomalies is determined. In the non-abelian case
it is shown to be compatible with the Wess“Zumino consistency conditions. The
relation of anomalies to the index of the Dirac operator in a gauge background is
discussed. Instantons are shown to contribute to the anomaly in CP(N-1) mod-
els and SU(2) gauge theories. The implications on the strong CP problem and
the U(1) problem are mentioned. While the study of anomalies has been limited
to the framework of perturbation theory for years, the lecture addresses also
recent breakthroughs in lattice ¬eld theory that allow non-perturbative investi-
gations of chiral anomalies. In particular, the overlap and domain wall fermion
formulations are described in detail, where lessons on supersymmetric quantum
mechanics and a two-dimensional model of a Dirac fermion in the background of
a static soliton help to illustrate the general idea behind domain wall fermions.
The lecture of Misha Shifman is devoted to “Supersymmetric Solitons and
Topology” and, in particular, on critical or BPS-saturated kinks and domain
walls. His discussion includes minimal N = 1 supersymmetric models of the
Landau“Ginzburg type in 1+1 dimensions, the minimal Wess“Zumino model
in 3+1 dimensions, and the supersymmetric CP(1) model in 1+1 dimensions,
which is a hybrid model (Landau“Ginzburg model on curved target space) that
possesses extended N = 2 supersymmetry. One of the main subjects of this
lecture is the variety of novel physical phenomena inherent to BPS-saturated
solitons in the presence of fermions. For example, the phenomenon of multiplet
shortening is described together with its implications on quantum corrections
to the mass (or wall tension) of the soliton. Moreover, irrationalization of the
4 E. Bick and F.D. Ste¬en

U(1) charge of the soliton is derived as an intriguing dynamical phenomena of
the N = 2 supersymmetric model with a topological term. The appendix of this
lecture presents an elementary introduction to supersymmetry, which emphasizes
its promises with respect to the problem of the cosmological constant and the
hierarchy problem.
The high hopes that supersymmetry, as a crucial basis of string theory, is a
key to a quantum theory of gravity and, thus, to the theory of everything must
be confronted with still missing experimental evidence for such a boson“fermion
symmetry. This demonstrates the importance of alternative approaches not rely-
ing on supersymmetry. A non-supersymmetric approach based on Connes™ non-
commutative geometry is presented by Thomas Sch¨cker in his lecture “Forces
from Connes™ geometry.” This lecture starts with a brief review of Einstein™s
derivation of general relativity from Riemannian geometry. Also the standard
model of particle physics is carefully reviewed with emphasis on its mathemat-
ical structure. Connes™ noncommutative geometry is illustrated by introducing
the reader step by step to Connes™ spectral triple. Einstein™s derivation of general
relativity is paralled in Connes™ language of spectral triples as a commutative
example. Here the Dirac operator de¬nes both the dynamics of matter and the
kinematics of gravity. A noncommutative example shows explicitly how a Yang“
Mills“Higgs model arises from gravity on a noncommutative geometry. The non-
commutative formulation of the standard model of particle physics is presented
and consequences for physics beyond the standard model are addressed. The
present status of this approach is described with a look at its promises towards
a uni¬cation of gravity with quantum ¬eld theory and at its open questions
concerning, for example, the construction of quantum ¬elds in noncommutative
space or spectral triples with Lorentzian signature. The appendix of this lecture
provides the reader with a compact review of the crucial mathematical basics
and de¬nitions used in this lecture.

3 Complementary Literature
Let us conclude this introduction with a brief guide to complementary literature
the reader might ¬nd useful. Further recommendations will be given in the lec-
tures. For quantum ¬eld theory, we appreciate very much the books of Peskin
and Schr¨der [1], Weinberg [2], and Zinn-Justin [3]. For general relativity, the
books of Wald [4] and Weinberg [5] can be recommended. More speci¬c texts we
found helpful in the study of topological aspects of quantum ¬eld theory are the
ones by Bertlmann [6], Coleman [7], Forkel [8], and Rajaraman [9]. For elabo-
rate treatments of the mathematical concepts, we refer the reader to the texts of
G¨ckeler and Sch¨cker [10], Nakahara [11], Nash and Sen [12], and Schutz [13].
o u

1. M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory
(Westview Press, Boulder 1995)
Introduction and Overview 5

2. S. Weinberg, The Quantum Theory Of Fields, Vols. I, II, and III, (Cambridge
University Press, Cambridge 1995, 1996, and 2000)
3. J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, 4th edn. (Caren-
don Press, Oxford 2002)
4. R. Wald, General Relativity (The University of Chicago Press, Chicago 1984)
5. S. Weinberg, Gravitation and Cosmology (Wiley, New York 1972)
6. R. A. Bertlmann, Anomalies in Quantum Field Theory (Oxford University Press,
Oxford 1996)
7. S. Coleman, Aspects of Symmetry (Cambridge University Press, Cambridge 1985)
8. H. Forkel, A Primer on Instantons in QCD, arXiv:hep-ph/0009136
9. R. Rajaraman, Solitons and Instantons (North-Holland, Amsterdam 1982)
10. M. G¨ckeler and T. Sch¨cker, Di¬erential Geometry, Gauge Theories, and Gravity
o u
(Cambridge University Press, Cambridge 1987)
11. M. Nakahara, Geometry, Topology and Physics, 2nd ed. (IOP Publishing, Bristol
12. C. Nash and S. Sen, Topology and Geometry for Physicists (Academic Press, Lon-
don 1983)
13. B. F. Schutz, Geometrical Methods of Mathematical Physics (Cambridge University
Press, Cambridge 1980)
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Topological Concepts in Gauge Theories

F. Lenz

Institute for Theoretical Physics III, University of Erlangen-N¨ rnberg,
Staudstrasse 7, 91058 Erlangen, Germany

Abstract. In these lecture notes, an introduction to topological concepts and meth-
ods in studies of gauge ¬eld theories is presented. The three paradigms of topological
objects, the Nielsen“Olesen vortex of the abelian Higgs model, the ™t Hooft“Polyakov
monopole of the non-abelian Higgs model and the instanton of Yang“Mills theory,
are discussed. The common formal elements in their construction are emphasized and
their di¬erent dynamical roles are exposed. The discussion of applications of topological
methods to Quantum Chromodynamics focuses on con¬nement. An account is given
of various attempts to relate this phenomenon to topological properties of Yang“Mills
theory. The lecture notes also include an introduction to the underlying concept of
homotopy with applications from various areas of physics.

1 Introduction
In a fragment [1] written in the year 1833, C. F. Gauß describes a profound
topological result which he derived from the analysis of a physical problem. He
considers the work Wm done by transporting a magnetic monopole (ein Ele-
ment des “positiven n¨rdlichen magnetischen Fluidums”) with magnetic charge
g along a closed path C1 in the magnetic ¬eld B generated by a current I ¬‚owing
along a closed loop C2 . According to the law of Biot“Savart, Wm is given by
I lk{C1 , C2 }.
Wm = g B(s1 ) ds1 =

Gauß recognized that Wm neither depends on the geometrical details of the
current carrying loop C2 nor on those of the closed path C1 .
(ds1 — ds2 ) · s12
lk{C1 , C2 } = (1)
|s12 |3
4π C1 C2

s12 = s2 ’ s1

Fig. 1. Transport of a magnetic charge along C1 in the magnetic ¬eld generated by a
current ¬‚owing along C2
Under continuous deformations of these curves, the value of lk{C1 , C2 }, the Link-
ing Number (“Anzahl der Umschlingungen”), remains unchanged. This quantity
is a topological invariant. It is an integer which counts the (signed) number of

F. Lenz, Topological Concepts in Gauge Theories, Lect. Notes Phys. 659, 7“98 (2005)
http://www.springerlink.com/ c Springer-Verlag Berlin Heidelberg 2005
8 F. Lenz


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