Thus, the interaction energy of the magnetic monopoles grows linearly with their

separation. In Quantum Chromodynamics (QCD) one is looking for mechanisms

of con¬nement of (chromo-) electric charges. Thus one attempts to transfer this

mechanism by some “duality transformation” which interchanges the role of

electric and magnetic ¬elds and charges. In view of such applications to QCD, it

should be emphasized that formation of vortices does not happen spontaneously.

It requires a minimal value of the applied ¬eld which depends on the microscopic

structure of the material and varies over three orders of magnitude [13].

The point κ = β = 1 in the parameter space of the abelian Higgs model

is very special. It separates Type I from Type II superconductors. I will now

show that at this point the energy of a vortex is determined by its charge. To

this end, I ¬rst derive a bound on the energy of the topological excitations, the

“Bogomol™nyi bound” [14]. Via an integration by parts, the energy (25) can be

written in the following form

E 1

d2 x [B ± (φφ— ’ 1)]

2 2

d2 x |[(‚x ’ iAx ) ± i(‚y ’ iAy )] φ| +

=

a2 2

1

d2 x [φ— φ ’ 1]

2

± d2 xB + (β ’ 1)

2

with the sign chosen according to the sign of the winding number n (cf. (22)).

For “critical coupling” β = 1 (cf. (24)), the energy is bounded by the third term

on the right-hand side, which in turn is given by the winding number (22)

E ≥ 2π|n| .

The Bogomol™nyi bound is saturated if the vortex satis¬es the following ¬rst

order di¬erential equations

[(‚x ’ iAx ) ± i(‚y ’ iAy )] φ = 0

Topological Concepts in Gauge Theories 19

B = ±(φφ— ’ 1) .

It can be shown that for β = 1 this coupled system of ¬rst order di¬erential

equations is equivalent to the Euler“Lagrange equations. The energy of these

particular solutions to the classical ¬eld equations is given in terms of the mag-

netic charge. Neither the existence of solutions whose energy is determined by

topological properties, nor the reduction of the equations of motion to a ¬rst or-

der system of di¬erential equations is a peculiar property of the Nielsen“Olesen

vortices. We will encounter again the Bogomol™nyi bound and its saturation in

our discussion of the ™t Hooft monopole and of the instantons. Similar solu-

tions with the energy determined by some charge play also an important role in

supersymmetric theories and in string theory.

A wealth of further results concerning the topological excitations in the

abelian Higgs model has been obtained. Multi-vortex solutions, ¬‚uctuations

around spherically symmetric solutions, supersymmetric extensions, or exten-

sions to non-commutative spaces have been studied. Finally, one can introduce

fermions by a Yukawa coupling

¯ ¯ /ψ

δL ∼ gφψψ + eψA

to the scalar and a minimal coupling to the Higgs ¬eld. Again one ¬nds what

will turn out to be a quite general property. Vortices induce fermionic zero

modes [15,16]. We will discuss this phenomenon in the context of instantons.

3 Homotopy

3.1 The Fundamental Group

In this section I will describe extensions and generalizations of the rather intuitive

concepts which have been used in the analysis of the abelian Higgs model. From

the physics point of view, the vacuum degeneracy is the essential property of

the abelian Higgs model which ultimately gives rise to the quantization of the

magnetic ¬‚ux and the emergence of topological excitations. More formally, one

views ¬elds like the Higgs ¬eld as providing a mapping of the asymptotic circle

in con¬guration space to the space of zeroes of the Higgs potential. In this way,

the quantization is a consequence of the presence of integer valued topological

invariants associated with this mapping. While in the abelian Higgs model these

properties are almost self-evident, in the forthcoming applications the structure

of the spaces to be mapped is more complicated. In the non-abelian Higgs model,

for instance, the space of zeroes of the Higgs potential will be a subset of a

non-abelian group. In such situations, more advanced mathematical tools have

proven to be helpful for carrying out the analysis. In our discussion and for

later applications, the concept of homotopy will be central (cf. [17,18]). It is

a concept which is relevant for the characterization of global rather than local

properties of spaces and maps (i.e. ¬elds). In the following we will assume that

the spaces are “topological spaces”, i.e. sets in which open subsets with certain

20 F. Lenz

properties are de¬ned and thereby the concept of continuity (“smooth maps”)

can be introduced (cf. [19]). In physics, one often requires di¬erentiability of

functions. In this case, the topological spaces must possess additional properties

(di¬erentiable manifolds). We start with the formal de¬nition of homotopy.

De¬nition: Let X, Y be smooth manifolds and f : X ’ Y a smooth map

between them. A homotopy or deformation of the map f is a smooth map

F :X —I ’Y (I = [0, 1])

with the property

F (x, 0) = f (x)

Each of the maps ft (x) = F (x, t) is said to be homotopic to the initial map

f0 = f and the map of the whole cylinder X —I is called a homotopy. The relation

of homotopy between maps is an equivalence relation and therefore allows to

divide the set of smooth maps X ’ Y into equivalence classes, homotopy classes.

De¬nition: Two maps f, g are called homotopic, f ∼ g, if they can be deformed

continuously into each other.

The mappings

Rn ’ Rn : f (x) = x, g(x) = x0 = const.

are homotopic with the homotopy given by

F (x, t) = (1 ’ t)x + tx0 . (32)

Spaces X in which the identity mapping 1X and the constant mapping are

homotopic, are homotopically equivalent to a point. They are called contractible.

De¬nition: Spaces X and Y are de¬ned to be homotopically equivalent if con-

tinuous mappings exist

f :X’Y g:Y ’X

,

such that

g —¦ f ∼ 1X f —¦ g ∼ 1Y

,

An important example is the equivalence of the n’sphere and the punctured

Rn+1 (one point removed)

S n = {x ∈ Rn+1 |x2 + x2 + . . . + x2 = 1} ∼ Rn+1 \{0}. (33)

1 2 n+1

which can be proved by stereographic projection. It shows that with regard

to homotopy, the essential property of a circle is the hole inside. Topologically

identical (homeomorphic) spaces, i.e. spaces which can be mapped continuously

and bijectively onto each other, possess the same connectedness properties and

are therefore homotopically equivalent. The converse is not true.

In physics, we often can identify the parameter t as time. Classical ¬elds,

evolving continuously in time are examples of homotopies. Here the restriction to

Topological Concepts in Gauge Theories 21

Fig. 4. Phase of matter ¬eld with winding number n = 0

continuous functions follows from energy considerations. Discontinuous changes

of ¬elds are in general connected with in¬nite energies or energy densities. For

instance, a homotopy of the “spin system” shown in Fig. 4 is provided by a

spin wave connecting some initial F (x, 0) with some ¬nal con¬guration F (x, 1).

Homotopy theory classi¬es the di¬erent sectors (equivalence classes) of ¬eld con-

¬gurations. Fields of a given sector can evolve into each other as a function of

time. One might be interested, whether the con¬guration of spins in Fig. 3 can

evolve with time from the ground state con¬guration shown in Fig. 4.

The Fundamental Group. The fundamental group characterizes connected-

ness properties of spaces related to properties of loops in these spaces. The basic

idea is to detect defects “ like a hole in the plane “ by letting loops shrink to

a point. Certain defects will provide a topological obstruction to such attempts.

Here one considers arcwise (or path) connected spaces, i.e. spaces where any pair

of points can be connected by some path.

A loop (closed path) through x0 in M is formally de¬ned as a map

± : [0, 1] ’ M with ±(0) = ±(1) = x0 .

A product of two loops is de¬ned by

±

1

±(2t) 0¤t¤

,

γ =±—β, 2,

γ(t) =

β(2t ’ 1) , ¤t¤1

1

2

and corresponds to traversing the loops consecutively. Inverse and constant loops

are given by

±’1 (t) = ±(1 ’ t), 0¤t¤1

and

c(t) = x0

respectively. The inverse corresponds to traversing a given loop in the opposite

direction.

De¬nition: Two loops through x0 ∈ M are said to be homotopic, ± ∼ β, if

they can be continuously deformed into each other, i.e. if a mapping H exists,

H : [0, 1] — [0, 1] ’ M ,

22 F. Lenz

with the properties

H(s, 0) = ±(s), 0 ¤ s ¤ 1 ; H(s, 1) = β(s),

H(0, t) = H(1, t) = x0 , 0 ¤ t ¤ 1. (34)

Once more, we may interpret t as time and the homotopy H as a time-dependent

evolution of loops into each other.

De¬nition: π1 (M, x0 ) denotes the set of equivalence classes (homotopy classes)

of loops through x0 ∈ M .

The product of equivalence classes is de¬ned by the product of their rep-

resentatives. It can be easily seen that this de¬nition does not depend on the

loop chosen to represent a certain class. In this way, π1 (M, x0 ) acquires a group

structure with the constant loop representing the neutral element. Finally, in an

arcwise connected space M , the equivalence classes π1 (M, x0 ) are independent of

the base point x0 and one therefore denotes with π1 (M ) the fundamental group

of M .

For applications, it is important that the fundamental group (or more gener-

ally the homotopy groups) of homotopically equivalent spaces X, Y are identical

π1 (X) = π1 (Y ).

Examples and Applications. Trivial topological spaces as far as their con-

nectedness is concerned are simply connected spaces.

De¬nition: A topological space X is said to be simply connected if any loop in

X can be continuously shrunk to a point.

The set of equivalence classes consists of one element, represented by the

constant loop and one writes

π1 = 0.

Obvious examples are the spaces Rn .

Non-trivial connectedness properties are the source of the peculiar properties

of the abelian Higgs model. The phase of the Higgs ¬eld θ de¬ned on a loop at

in¬nity, which can continuously be deformed into a circle at in¬nity, de¬nes a

mapping

θ : S1 ’ S1.

An arbitrary phase χ de¬ned on S 1 has the properties

χ(0) = 0 , χ(2π) = 2πm . (35)

It can be continuously deformed into the linear function m•. The mapping

χ(2π)

H(•, t) = (1 ’ t) χ(•) + t •

2π

with the properties

H(0, t) = χ(0) = 0 , H(2π, t) = χ(2π) ,

Topological Concepts in Gauge Theories 23

is a homotopy and thus

χ(•) ∼ m•.

The equivalence classes are therefore characterized by integers m and since these

winding numbers are additive when traversing two loops

π1 (S 1 ) ∼ Z. (36)

Vortices are de¬ned on R2 \{0} since the center of the vortex, where θ(x) is

ill-de¬ned, has to be removed. The homotopic equivalence of this space to S 1

(33) implies that a vortex with winding number N = 0 is stable; it cannot evolve

with time into the homotopy class of the ground-state con¬guration where up to

continuous deformations, the phase points everywhere into the same direction.

This argument also shows that the (abelian) vortex is not topologically stable

in higher dimensions. In Rn \{0} with n ≥ 3, by continuous deformation, a loop

can always avoid the origin and can therefore be shrunk to a point. Thus

π1 (S n ) = 0 , n ≥ 2 , (37)

i.e. n’spheres with n > 1 are simply connected. In particular, in 3 dimensions a

“point defect” cannot be detected by the fundamental group. On the other hand,

if we remove a line from the R3 , the fundamental group is again characterized

by the winding number and we have

π1 (R3 \R) ∼ Z . (38)

This result can also be seen as a consequence of the general homotopic equiva-

lence

Rn+1 \R ∼ S n’1 . (39)

The result (37) implies that stringlike objects in 3-dimensional spaces can be

detected by loops and that their topological stability is determined by the non-

triviality of the fundamental group. For constructing pointlike objects in higher

dimensions, the ¬elds must assume values in spaces with di¬erent connectedness

properties.

The fundamental group of a product of spaces X, Y is isomorphic to the

product of their fundamental groups

π1 (X — Y ) ∼ π1 (X) — π1 (Y ) . (40)

For a torus T and a cylinder C we thus have