intersections of the loop C1 with an arbitrary (oriented) surface in R3 whose

boundary is the loop C2 (cf. [2,3]). In the same note, Gauß deplores the lit-

tle progress in topology (“Geometria Situs”) since Leibniz™s times who in 1679

postulated “another analysis, purely geometric or linear which also de¬nes the

position (situs), as algebra de¬nes magnitude”. Leibniz also had in mind appli-

cations of this new branch of mathematics to physics. His attempt to interest a

physicist (Christiaan Huygens) in his ideas about topology however was unsuc-

cessful. Topological arguments made their entrance in physics with the formula-

tion of the Helmholtz laws of vortex motion (1858) and the circulation theorem

by Kelvin (1869) and until today hydrodynamics continues to be a fertile ¬eld

for the development and applications of topological methods in physics. The

success of the topological arguments led Kelvin to seek for a description of the

constituents of matter, the atoms at that time in terms of vortices and thereby

explain topologically their stability. Although this attempt of a topological ex-

planation of the laws of fundamental physics, the ¬rst of many to come, had to

fail, a classi¬cation of knots and links by P. Tait derived from these e¬orts [4].

Today, the use of topological methods in the analysis of properties of sys-

tems is widespread in physics. Quantum mechanical phenomena such as the

Aharonov“Bohm e¬ect or Berry™s phase are of topological origin, as is the sta-

bility of defects in condensed matter systems, quantum liquids or in cosmology.

By their very nature, topological methods are insensitive to details of the systems

in question. Their application therefore often reveals unexpected links between

seemingly very di¬erent phenomena. This common basis in the theoretical de-

scription not only refers to obvious topological objects like vortices, which are

encountered on almost all scales in physics, it applies also to more abstract

concepts. “Helicity”, for instance, a topological invariant in inviscid ¬‚uids, dis-

covered in 1969 [5], is closely related to the topological charge in gauge theories.

Defects in nematic liquid crystals are close relatives to defects in certain gauge

theories. Dirac™s work on magnetic monopoles [6] heralded in 1931 the relevance

of topology for ¬eld theoretic studies in physics, but it was not until the for-

mulation of non-abelian gauge theories [7] with their wealth of non-perturbative

phenomena that topological methods became a common tool in ¬eld theoretic

investigations.

In these lecture notes, I will give an introduction to topological methods in

gauge theories. I will describe excitations with non-trivial topological properties

in the abelian and non-abelian Higgs model and in Yang“Mills theory. The topo-

logical objects to be discussed are instantons, monopoles, and vortices which in

space-time are respectively singular on a point, a world-line, or a world-sheet.

They are solutions to classical non-linear ¬eld equations. I will emphasize both

their common formal properties and their relevance in physics. The topologi-

cal investigations of these ¬eld theoretic models is based on the mathematical

concept of homotopy. These lecture notes include an introductory section on ho-

motopy with emphasis on applications. In general, proofs are omitted or replaced

by plausibility arguments or illustrative examples from physics or geometry. To

emphasize the universal character in the topological analysis of physical sys-

tems, I will at various instances display the often amazing connections between

Topological Concepts in Gauge Theories 9

very di¬erent physical phenomena which emerge from such analyses. Beyond the

description of the paradigms of topological objects in gauge theories, these lec-

ture notes contain an introduction to recent applications of topological methods

to Quantum Chromodynamics with emphasis on the con¬nement issue. Con-

¬nement of the elementary degrees of freedom is the trademark of Yang“Mills

theories. It is a non-perturbative phenomenon, i.e. the non-linearity of the the-

ory is as crucial here as in the formation of topologically non-trivial excitations.

I will describe various ideas and ongoing attempts towards a topological charac-

terization of this peculiar property.

2 Nielsen“Olesen Vortex

The Nielsen“Olesen vortex [8] is a topological excitation in the abelian Higgs

model. With topological excitation I will denote in the following a solution to the

¬eld equations with non-trivial topological properties. As in all the subsequent

examples, the Nielsen“Olesen vortex owes its existence to vacuum degeneracy,

i.e. to the presence of multiple, energetically degenerate solutions of minimal

energy. I will start with a brief discussion of the abelian Higgs model and its

(classical) “ground states”, i.e. the ¬eld con¬gurations with minimal energy.

2.1 Abelian Higgs Model

The abelian Higgs Model is a ¬eld theoretic model with important applications

in particle and condensed matter physics. It constitutes an appropriate ¬eld

theoretic framework for the description of phenomena related to superconduc-

tivity (cf. [9,10]) (“Ginzburg“Landau Model”) and its topological excitations

(“Abrikosov-Vortices”). At the same time, it provides the simplest setting for

the mechanism of mass generation operative in the electro-weak interaction.

The abelian Higgs model is a gauge theory. Besides the electromagnetic ¬eld

it contains a self-interacting scalar ¬eld (Higgs ¬eld) minimally coupled to elec-

tromagnetism. From the conceptual point of view, it is advantageous to consider

this ¬eld theory in 2 + 1 dimensional space-time and to extend it subsequently

to 3 + 1 dimensions for applications.

The abelian Higgs model Lagrangian

1

L = ’ Fµν F µν + (Dµ φ)— (Dµ φ) ’ V (φ) (2)

4

contains the complex (charged), self-interacting scalar ¬eld φ. The Higgs poten-

tial

1

V (φ) = »(|φ|2 ’ a2 )2 . (3)

4

as a function of the real and imaginary part of the Higgs ¬eld is shown in Fig. 2.

By construction, this Higgs potential is minimal along a circle |φ| = a in the

complex φ plane. The constant » controls the strength of the self-interaction of

the Higgs ¬eld and, for stability reasons, is assumed to be positive

» ≥ 0. (4)

10 F. Lenz

Fig. 2. Higgs Potential V (φ)

The Higgs ¬eld is minimally coupled to the radiation ¬eld Aµ , i.e. the partial

derivative ‚µ is replaced by the covariant derivative

Dµ = ‚µ + ieAµ . (5)

Gauge ¬elds and ¬eld strengths are related by

1

Fµν = ‚µ Aν ’ ‚ν Aµ = [Dµ , Dν ] .

ie

Equations of Motion

• The (inhomogeneous) Maxwell equations are obtained from the principle of

least action,

d4 xL = 0 ,

δS = δ

by variation of S with respect to the gauge ¬elds. With

δL δL

= ’F µν , = ’j ν ,

δ‚µ Aν δAν

we obtain

jν = ie(φ ‚ν φ ’ φ‚ν φ ) ’ 2e2 φ— φAν .

‚µ F µν = j ν ,

• The homogeneous Maxwell equations are not dynamical equations of mo-

tion “ they are integrability conditions and guarantee that the ¬eld strength

can be expressed in terms of the gauge ¬elds. The homogeneous equations

follow from the Jacobi identity of the covariant derivative

[Dµ , [Dν , Dσ ]] + [Dσ , [Dµ , Dν ]] + [Dν , [Dσ , Dµ ]] = 0.

µνρσ

Multiplication with the totally antisymmetric tensor, , yields the ho-

˜

mogeneous equations for the dual ¬eld strength F µν

1

˜ ˜

Dµ , F µν = 0 , F µν = µνρσ

Fρσ .

2

Topological Concepts in Gauge Theories 11

The transition

F ’F

˜

corresponds to the following duality relation of electric and magnetic ¬elds

E’B B ’ ’E.

,

• Variation with respect to the charged matter ¬eld yields the equation of

motion

δV

Dµ Dµ φ + — = 0.

δφ

Gauge theories contain redundant variables. This redundancy manifests itself in

the presence of local symmetry transformations; these “gauge transformations”

U (x) = eie±(x) (6)

rotate the phase of the matter ¬eld and shift the value of the gauge ¬eld in a

space-time dependent manner

1

‚µ U † (x) .

φ ’ φ [U ] = U (x)φ(x) , Aµ ’ Aµ ] = Aµ + U (x)

[U

(7)

ie

The covariant derivative Dµ has been de¬ned such that Dµ φ transforms co-

variantly, i.e. like the matter ¬eld φ itself.

Dµ φ(x) ’ U (x) Dµ φ(x).

This transformation property together with the invariance of Fµν guarantees

invariance of L and of the equations of motion. A gauge ¬eld which is gauge

equivalent to Aµ = 0 is called a pure gauge. According to (7) a pure gauge

satis¬es

1

Apg (x) = U (x) ‚µ U † (x) = ’‚µ ±(x) , (8)

µ

ie

and the corresponding ¬eld strength vanishes.

Canonical Formalism. In the canonical formalism, electric and magnetic ¬elds

play distinctive dynamical roles. They are given in terms of the ¬eld strength

tensor by

1

E i = ’F 0i , B i = ’ ijk Fjk = (rotA)i .

2

Accordingly,

1 1

’ Fµν F µν = E2 ’ B2 .

4 2

The presence of redundant variables complicates the formulation of the canon-

ical formalism and the quantization. Only for independent dynamical degrees

of freedom canonically conjugate variables may be de¬ned and corresponding

commutation relations may be associated. In a ¬rst step, one has to choose by a

“gauge condition” a set of variables which are independent. For the development

12 F. Lenz

of the canonical formalism there is a particularly suited gauge, the “Weyl” “ or

“temporal” gauge

A0 = 0. (9)

We observe, that the time derivative of A0 does not appear in L, a property

which follows from the antisymmetry of the ¬eld strength tensor and is shared

by all gauge theories. Therefore in the canonical formalism A0 is a constrained

variable and its elimination greatly simpli¬es the formulation. It is easily seen

that (9) is a legitimate gauge condition, i.e. that for an arbitrary gauge ¬eld a

gauge transformation (7) with gauge function

‚0 ±(x) = A0 (x)

indeed eliminates A0 . With this gauge choice one proceeds straightforwardly

with the de¬nition of the canonically conjugate momenta

δL δL

= ’E i , = π,

δ‚0 Ai δ‚0 φ

and constructs via Legendre transformation the Hamiltonian density

12

(E + B 2 ) + π — π + (Dφ)— (Dφ) + V (φ) ,

H= d3 xH(x) .

H= (10)

2

With the Hamiltonian density given by a sum of positive de¬nite terms (cf.(4)),

the energy density of the ¬elds of lowest energy must vanish identically. There-

fore, such ¬elds are static

E = 0, π = 0, (11)

with vanishing magnetic ¬eld

B = 0. (12)

The following choice of the Higgs ¬eld

|φ| = a, i.e. φ = aeiβ (13)

renders the potential energy minimal. The ground state is not unique. Rather

the system exhibits a “vacuum degeneracy”, i.e. it possesses a continuum of ¬eld

con¬gurations of minimal energy. It is important to characterize the degree of

this degeneracy. We read o¬ from (13) that the manifold of ¬eld con¬gurations

of minimal energy is given by the manifold of zeroes of the potential energy. It

is characterized by β and thus this manifold has the topological properties of a

circle S 1 . As in other examples to be discussed, this vacuum degeneracy is the

source of the non-trivial topological properties of the abelian Higgs model.

To exhibit the physical properties of the system and to study the conse-

quences of the vacuum degeneracy, we simplify the description by performing

a time independent gauge transformation. Time independent gauge transforma-

tions do not alter the gauge condition (9). In the Hamiltonian formalism, these

gauge transformations are implemented as canonical (unitary) transformations

Topological Concepts in Gauge Theories 13

which can be regarded as symmetry transformations. We introduce the modulus

and phase of the static Higgs ¬eld

φ(x) = ρ(x)eiθ(x) ,

and choose the gauge function

±(x) = ’θ(x) (14)

so that in the transformation (7) to the “unitary gauge” the phase of the matter

¬eld vanishes

1

A[U ] = A ’ ∇θ(x) , (Dφ)[U ] = ∇ρ(x) ’ ieA[U ] ρ(x) .

φ[U ] (x) = ρ(x) ,

e

This results in the following expression for the energy density of the static ¬elds