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

u

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

o

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

References

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

2003)

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,

u

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

o

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

4πg

I lk{C1 , C2 }.

Wm = g B(s1 ) ds1 =

c

C1

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

1

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