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2 L

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 •

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

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