will see later, under suitable assumptions concerning the asymptotic behavior of

gauge ¬elds, in Yang“Mills theories, each gauge orbit is labeled by a topological

invariant, the topological charge.

3.7 Defects in Ordered Media

In condensed matter physics, topological methods ¬nd important applications

in the investigations of properties of defects occurring in ordered media [27].

For applying topological arguments, one has to specify the topological space

X in which the ¬elds describing the degrees of freedom are de¬ned and the

topological space M (target space) of the values of the ¬elds. In condensed

matter physics the (classical) ¬elds ψ(x) are called the order parameter and M

correspondingly the order parameter space. A system of spins or directors may

be de¬ned on lines, planes or in the whole space, i.e. X = Rn with n = 1, 2

or 3. The ¬elds or order parameters describing spins are spatially varying unit

vectors with arbitrary orientations: M = S 2 or if restricted to a plane M = S 1 .

The target spaces of directors are the corresponding projective spaces RP n .

A defect is a point, a line or a surface on which the order parameter is ill-

de¬ned. The defects are de¬ned accordingly as point defects (monopoles), line

defects (vortices, disclinations), or surface defects (domain walls). Such defects

are topologically stable if they cannot be removed by a continuous change in

the order parameter. Discontinuous changes require in physical systems of e.g.

spin degrees of freedom substantial changes in a large number of the degrees

Topological Concepts in Gauge Theories 35

of freedom and therefore large energies. The existence of singularities alter the

topology of the space X. Point and line defects induce respectively the following

changes in the topology: X = R3 ’ R3 \{0} ∼ S 2 and X = R3 ’ R3 \R1 ∼ S 1 .

Homotopy provides the appropriate tools to study the stability of defects. To

this end, we proceed as in the abelian Higgs model and investigate the order

parameter on a circle or a 2-sphere su¬ciently far away from the defect. In this

way, the order parameter de¬nes a mapping ψ : S n ’ M and the stability of the

defects is guaranteed if the homotopy group πn (M ) is non-trivial. Alternatively

one may study the defects by removing from the space X the manifold on which

the order parameter becomes singular. The structure of the homotopy group has

important implications for the dynamics of the defects. If the asymptotic circle

encloses two defects, and if the homotopy group is abelian, than in a merger of

the two defects the resulting defect is speci¬ed by the sum of the two integers

characterizing the individual defects. In particular, winding numbers π1 (S 1 )

and monopole charges π2 (S 2 ) (cf. (43)) are additive.

I conclude this discussion by illustrating some of the results using the ex-

amples of magnetic systems represented by spins and nematic liquid crystals

represented by directors, i.e. spins with indistinguishable heads and tails (cf.

(46) and the following discussion). If 2-dimensional spins (M = S 1 ) or directors

(M = RP 1 ) live on a plane (X = R2 ), a defect is topologically stable. The punc-

tured plane obtained by the removal of the defect is homotopically equivalent

to a circle (33) and the topological stability follows from the non-trivial homo-

topy group π1 (S 1 ) for magnetic substances. The argument applies to nematic

substances as well since identi¬cation of antipodal points of a circle yields again

a circle

RP 1 ∼ S 1 .

On the other hand, a point defect in a system of 3-dimensional spins M = S 2

de¬ned on a plane X = R2 “ or equivalently a line defect in X = R3 “ is not

stable. Removal of the defect manifold generates once more a circle. The triviality

of π1 (S 2 ) (cf. (37)) shows that the defect can be continuously deformed into a

con¬guration where all the spins point into the same direction. On S 2 a loop can

always be shrunk to a point (cf. (37)). In nematic substances, there are stable

point and line defects for X = R2 and X = R3 , respectively, since

π1 (RP 2 ) = Z2 .

Non-shrinkable loops on RP 2 are obtained by connecting a given point on S 2

with its antipodal one. Because of the identi¬cation of antipodal points, the line

connecting the two points cannot be contracted to a point. In the identi¬cation,

this line on S 2 becomes a non-contractible loop on RP 2 . Contractible and non-

contractible loops on RP 2 are shown in Fig. 6 . This ¬gure also demonstrates that

connecting two antipodal points with two di¬erent lines produces a contractible

loop. Therefore the space of loops contains only two inequivalent classes. More

generally, one can show (cf. [19])

π1 (RP n ) = Z2 , n ≥ 2, (73)

36 F. Lenz

Q = Q1 Q

Q

P1

b

a

a

a

Q1

P P

P = P1

Fig. 6. The left ¬gure shows loops a, b which on RP 2 can (b) and cannot (a) be shrunk

to a point. The two ¬gures on the right demonstrate how two loops of the type a can

be shrunk to one point. By moving the point P 1 together with its antipodal point Q1

two shrinkable loops of the type b are generated

and (cf. [18])

n ≥ 2.

πn (RP m ) = πn (S m ) , (74)

Thus, in 3-dimensional nematic substances point defects (monopoles), also pres-

ent in magnetic substances, and line defects (disclinations), absent in magnetic

substances, exist. In Fig. 7 the topologically stable line defect is shown. The

circles around the defect are mapped by θ(•) = • into RP 2 . Only due to the

2

identi¬cation of the directions θ ∼ θ+π this mapping is continuous. For magnetic

substances, it would be discontinuous along the • = 0 axis.

Liquid crystals can be considered with regard to their underlying dynamics as

close relatives to some of the ¬elds of particle physics. They exhibit spontaneous

orientations, i.e. they form ordered media with respect to ˜internal™ degrees of

freedom not joined by formation of a crystalline structure. Their topologically

stable defects are also encountered in gauge theories as we will see later. Unlike

the ¬elds in particle physics, nematic substances can be manipulated and, by

their birefringence property, allow for a beautiful visualization of the structure

and dynamics of defects (for a thorough discussion of the physics of liquid crystals

and their defects (cf. [29,30]). These substances o¬er the opportunity to study on

a macroscopic level, emergence of monopoles and their dynamics. For instance,

by enclosing a water droplet in a nematic liquid drop, the boundary conditions on

the surface of the water droplet and on the surface of the nematic drop cooperate

to generate a monopole (hedgehog) structure which, as Fig. 8C demonstrates,

Fig. 7. Line defect in RP 2 . In addition to the directors also the integral curves are

shown

Topological Concepts in Gauge Theories 37

Fig. 8. Nematic drops (A) containing one (C) or more water droplets (B) (the ¬gure

is taken from [31]). The distance between the defects is about 5 µm

can be observed via its peculiar birefringence properties, as a four armed star

of alternating bright and dark regions. If more water droplets are dispersed in

a nematic drop, they form chains (Fig. 8A) which consist of the water droplets

alternating with hyperbolic defects of the nematic liquid (Fig. 8B). The non-

trivial topological properties stabilize these objects for as long as a couple of

weeks [31]. In all the examples considered so far, the relevant fundamental

groups were abelian. In nematic substances the “biaxial nematic phase” has been

identi¬ed (cf. [29]) which is characterized by a non-abelian fundamental group.

The elementary constituents of this phase can be thought of as rectangular boxes

rather than rods which, in this phase are aligned. Up to 180 —¦ rotations around

π

the 3 mutually perpendicular axes (Ri ), the orientation of such a box is speci¬ed

by an element of the rotation group SO(3). The order parameter space of such

a system is therefore given by

D2 = {&, R1 , R2 , R3 }.

π π π

M = SO(3)/D2 ,

By representing the rotations by elements of SU (2) (cf. (64)), the group D2 is

extended to the group of 8 elements, containing the Pauli matrices (50),

Q = {±&, ±„ 1 , ±„ 2 , ±„ 3 } ,

the group of quaternions. With the help of the identities (59) and (60), we derive

π1 (SO(3)/D2 ) ∼ π1 (SU (2)/Q) ∼ Q . (75)

In the last step it has been used that in a discrete group the connected component

of the identity contains the identity only.

The non-abelian nature of the fundamental group has been predicted to

have important physical consequences for the behavior of defects in the nematic

biaxial phase. This concerns in particular the coalescence of defects and the

possibility of entanglement of disclination lines [29].

38 F. Lenz

4 Yang“Mills Theory

In this introductory section I review concepts, de¬nitions, and basic properties

of gauge theories.

Gauge Fields. In non-abelian gauge theories, gauge ¬elds are matrix-valued

functions of space-time. In SU(N) gauge theories they can be represented by

the generators of the corresponding Lie algebra, i.e. gauge ¬elds and their color

components are related by

a

»

Aa (x)

Aµ (x) = , (76)

µ

2

where the color sum runs over the N 2 ’ 1 generators. The generators are her-

mitian, traceless N — N matrices whose commutation relations are speci¬ed by

the structure constants f abc

»a »b »c

= if abc .

,

22 2

The normalization is chosen as

» a »b 1

·

tr = δab .

22 2

Most of our applications will be concerned with SU (2) gauge theories; in this

case the generators are the Pauli matrices (50)

»a = „ a ,

with structure constants

f abc = abc

.

Covariant derivative, ¬eld strength tensor, and its color components are respec-

tively de¬ned by

Dµ = ‚µ + igAµ , (77)

1

Fµν = ‚µ Aa ’ ‚ν Aa ’ gf abc Ab Ac .

F µν = a

[Dµ , Dν ], (78)

ν µ µν

ig

The de¬nition of electric and magnetic ¬elds in terms of the ¬eld strength tensor

is the same as in electrodynamics

1

E ia (x) = ’F 0ia (x) B ia (x) = ’ ijk

F jka (x) .

, (79)

2

The dimensions of gauge ¬eld and ¬eld strength in 4 dimensional space-time are

’1 ’2

[A] = , [F ] =

Topological Concepts in Gauge Theories 39

and therefore in absence of a scale

xν

∼

Aa a

Mµν ,

µ

x2

a

with arbitrary constants Mµν . In general, the action associated with these ¬elds

exhibits infrared and ultraviolet logarithmic divergencies. In the following we

will discuss

• Yang“Mills Theories

Only gauge ¬elds are present. The Yang“Mills Lagrangian is

1 1 1

LY M = ’ F µνa Fµν = ’ tr {F µν Fµν } = (E2 ’ B2 ).

a

(80)

4 2 2

• Quantum Chromodynamics

QCD contains besides the gauge ¬elds (gluons), fermion ¬elds (quarks).

Quarks are in the fundamental representation, i.e. in SU(2) they are rep-

resented by 2-component color spinors. The QCD Lagrangian is (¬‚avor de-

pendences suppressed)

¯

LQCD = LY M + Lm , Lm = ψ (iγ µ Dµ ’ m) ψ, (81)

with the action of the covariant derivative on the quarks given by

(Dµ ψ)i = (‚µ δ ij + igAij ) ψ j i, j = 1 . . . N

µ

• Georgi“Glashow Model

In the Georgi“Glashow model [32] (non-abelian Higgs model), the gluons

are coupled to a scalar, self-interacting (V (φ)) (Higgs) ¬eld in the adjoint

representation. The Higgs ¬eld has the same representation in terms of the

generators as the gauge ¬eld (76) and can be thought of as a 3-component

color vector in SU (2). Lagrangian and action of the covariant derivative are

respectively

1

LGG = LY M + Lm , Lm = Dµ φDµ φ ’ V (φ) , (82)

2

= (‚µ δ ac ’ gf abc Ab )φc .

(Dµ φ)a = [Dµ , φ ] a