‚xβ ‚xβ

f ’1 (Ui ) Ui

i i

where the space is represented as a union of disjoint neighborhoods Ui with

y0 ∈ Ui and non-vanishing Jacobian.

3.5 Topological Groups

In many application of topological methods to physical systems, the relevant

degrees of freedom are described by ¬elds which take values in topological groups

like the Higgs ¬eld in the abelian or non-abelian Higgs model or link variables

and Wilson loops in gauge theories. In condensed matter physics an important

example is the order parameter in super¬‚uid 3 He in the “A-phase” in which the

pairing of the Helium atoms occurs in p-states with the spins coupled to 1. This

pairing mechanism is the source of a variety of di¬erent phenomena and gives

rise to the rather complicated manifold of the order parameter SO(3) — S 2 /Z2

(cf. [23]).

SU(2) as Topological Space. The group SU (2) of unitary transformations is

of fundamental importance for many applications in physics. It can be generated

by the Pauli-matrices

0 ’i

01 10

„1 = , „2 = , „3 = (50)

0 ’1 .

10 i0

Every element of SU (2) can be parameterized in the following way

ˆ

U = eiφ·„ = cos φ + i„ · φ sin φ = a + i„ · b. (51)

Here φ denotes an arbitrary vector in internal (e.g. isospin or color) space and

we do not explicitly write the neutral element e. This vector is parameterized

by the 4 (real) parameters a, b subject to the unitarity constraint

U U † = (a + ib · „ )(a ’ ib · „ ) = a2 + b2 = 1 .

This parameterization establishes the topological equivalence (homeomorphism)

of SU (2) and S 3

SU (2) ∼ S 3 . (52)

30 F. Lenz

This homeomorphism together with the results (43) and (44) shows

π3 SU (2) = Z.

π1,2 SU (2) = 0, (53)

One can show more generally the following properties of homotopy groups

πk SU (n) = 0 k < n .

The triviality of the fundamental group of SU (2) (53) can be veri¬ed by con-

structing an explicit homotopy between the loop

u2n (s) = exp{i2nπs„ 3 } (54)

and the constant map

uc (s) = 1 . (55)

The mapping

π π

’ i t„ 1 exp i t(„ 1 cos 2πns + „ 2 sin 2πns)

H(s, t) = exp

2 2

has the desired properties (cf. (34))

H(s, 0) = 1, H(s, 1) = u2n (s), H(0, t) = H(1, t) = 1 ,

as can be veri¬ed with the help of the identity (51). After continuous deforma-

tions and proper choice of the coordinates on the group manifold, any loop can

be parameterized in the form 54.

Not only Lie groups but also quotient spaces formed from them appear in

important physical applications. The presence of the group structure suggests

the following construction of quotient spaces. Given any subgroup H of a group

G, one de¬nes an equivalence between two arbitrary elements g1 , g2 ∈ G if they

are identical up to multiplication by elements of H

’1

g1 ∼ g2 i¬ g1 g2 ∈ H . (56)

The set of elements in G which are equivalent to g ∈ G is called the left coset

(modulo H) associated with g and is denoted by

g H = {gh |h ∈ H} . (57)

The space of cosets is called the coset space and denoted by

G/H = {gH |g ∈ G} . (58)

If N is an invariant or normal subgroup, i.e. if gN g ’1 = N for all g ∈ G, the

coset space is actually a group with the product de¬ned by (g1 N ) · (g2 N ) =

g1 g2 N . It is called the quotient or factor group of G by N .

As an example we consider the group of translations in R3 . Since this is an

abelian group, each subgroup is normal and can therefore be used to de¬ne factor

Topological Concepts in Gauge Theories 31

groups. Consider N = Tx the subgroup of translations in the x’direction. The

cosets are translations in the y-z plane followed by an arbitrary translation in the

x’direction. The factor group consists therefore of translations with unspeci¬ed

parameter for the translation in the x’direction. As a further example consider

rotations R (•) around a point in the x ’ y plane. The two elements

e = R (0) , r = R (π)

form a normal subgroup N with the factor group given by

G/N = {R (•) N |0 ¤ • < π} .

Homotopy groups of coset spaces can be calculated with the help of the following

two identities for connected and simply connected Lie-groups such as SU (n).

With H0 we denote the component of H which is connected to the neutral

element e. This component of H is an invariant subgroup of H. To verify this,

denote with γ(t) the continuous curve which connects the unity e at t = 0 with

an arbitrary element h0 of H0 . With γ(t) also hγ(t)h’1 is part of H0 for arbitrary

h ∈ H. Thus H0 is a normal subgroup of H and the coset space H/H0 is a group.

One extends the de¬nition of the homotopy groups and de¬nes

π0 (H) = H/H0 . (59)

The following identities hold (cf. [24,25])

π1 (G/H) = π0 (H) , (60)

and

π2 (G/H) = π1 (H0 ) . (61)

Applications of these identities to coset spaces of SU (2) will be important in the

following. We ¬rst observe that, according to the parameterization (51), together

with the neutral element e also ’e is an element of SU (2) φ = 0, π in (51) .

These 2 elements commute with all elements of SU(2) and form a subgroup , the

center of SU (2)

Z SU (2) = { e, ’e } ∼ Z2 . (62)

According to the identity (60) the fundamental group of the factor group is

non-trivial

π1 SU (2)/Z(SU (2)) = Z2 . (63)

As one can see from the following argument, this result implies that the group of

rotations in 3 dimensions SO(3) is not simply connected. Every rotation matrix

Rij ∈ SO(3) can be represented in terms of SU (2) matrices (51)

1

tr U „ i U † „ j .

Rij [U ] =

4

The SU (2) matrices U and ’U represent the same SO(3) matrix. Therefore,

SO(3) ∼ SU (2)/Z2 (64)

32 F. Lenz

and thus

π1 SO(3) = Z2 , (65)

i.e. SO(3) is not simply connected.

We have veri¬ed above that the loops u2n (s) (54) can be shrunk in SU(2) to

a point. They also can be shrunk to a point on SU (2)/Z2 . The loop

u1 (s) = exp{iπs„ 3 } (66)

connecting antipodal points however is topologically stable on SU (2)/Z2 , i.e. it

cannot be deformed continuously to a point, while its square, u2 (s) = u2 (s) can

1

be.

The identity (61) is important for the spontaneous symmetry breakdown

with a remaining U (1) gauge symmetry. Since the groups SU (n) are simply

connected, one obtains

π2 SU (n)/U (1) = Z. (67)

3.6 Transformation Groups

Historically, groups arose as collections of permutations or one-to-one transfor-

mations of a set X onto itself with composition of mappings as the group product.

If X contains just n elements, the collection S (X) of all its permutations is the

symmetric group with n! elements. In F. Klein™s approach, to each geometry is

associated a group of transformations of the underlying space of the geometry.

For example, the group E(2) of Euclidean plane geometry is the subgroup of

S E 2 which leaves the distance d (x, y) between two arbitrary points in the

plane (E 2 ) invariant, i.e. a transformation

T : E2 ’ E2

is in the group i¬

d (T x, T y) = d (x, y) .

The group E(2) is also called the group of rigid motions. It is generated by

translations, rotations, and re¬‚ections. Similarly, the general Lorentz group is the

group of Poincar´ transformations which leave the (relativistic) distance between

e

two space-time points invariant. The interpretation of groups as transformation

groups is very important in physics. Mathematically, transformation groups are

de¬ned in the following way (cf.[26]):

De¬nition: A Lie group G is represented as a group of transformations of a

manifold X (left action on X) if there is associated with each g ∈ G a di¬eomor-

phism of X to itself

x ’ Tg (x) , x∈X with Tg1 g2 = Tg1 Tg2

(“right action” Tg1 g2 = Tg2 Tg1 ) and if Tg (x) depends smoothly on the arguments

g, x.

Topological Concepts in Gauge Theories 33

If G is any of the Lie groups GL (n, R) , O (n, R) , GL (n, C) , U (n) then G

acts in the obvious way on the manifold Rn or C n .

The orbit of x ∈ X is the set

Gx = {Tg (x) |g ∈ G} ‚ X . (68)

The action of a group G on a manifold X is said to be transitive if for every

two points x, y ∈ X there exists g ∈ G such that Tg (x) = y, i.e. if the orbits

satisfy Gx = X for every x ∈ X . Such a manifold is called a homogeneous

space of the Lie group. The prime example of a homogeneous space is R3 under

translations; every two points can be connected by translations. Similarly, the

n

group of translations acts transitively on the n’dimensional torus T n = S 1

in the following way:

Ty (z) = e2iπ(•1 +t1 ) , ..., e2iπ(•n +tn )

with

y = (t1 , ..., tn ) ∈ Rn , z = e2iπ(•1 ) , ..., e2iπ(•n ) ∈ T n .

If the translations are given in terms of integers, ti = ni , we have Tn (z) = z.

This is a subgroup of the translations and is de¬ned more generally:

De¬nition: The isotropy group Hx of the point x ∈ X is the subgroup of all

elements of G leaving x ¬xed and is de¬ned by

Hx = {g ∈ G|Tg (x) = x} . (69)

The group O (n + 1) acts transitively on the sphere S n and thus S n is a

homogeneous space for the Lie group O (n + 1) of orthogonal transformations of

Rn+1 . The isotropy group of the point x = (1, 0, ...0) ∈ S n is comprised of all

matrices of the form

10

, A ∈ O (n)

0A

describing rotations around the x1 axis.

Given a transformation group G acting on a manifold X, we de¬ne orbits as

the equivalence classes, i.e.

x∼y if for some g ∈ G y = g x.

For X = Rn and G = O(n) the orbits are concentric spheres and thus in one

to one correspondence with real numbers r ≥ 0. This is a homeomorphism of

Rn /O (n) on the ray 0 ¤ r ¤ ∞ (which is almost a manifold).

If one de¬nes points on S 2 to be equivalent if they are connected by a rotation

around a ¬xed axis, the z axis, the resulting quotient space S 2 /O(2) consists of

all the points on S 2 with ¬xed azimuthal angle, i.e. the quotient space is a

segment

S 2 /O(2) = {θ | 0 ¤ θ ¤ π} . (70)

34 F. Lenz

Note that in the integration over the coset spaces Rn /O(n) and S 2 /O(2) the

radial volume element rn’1 and the volume element of the polar angle sin θ

appear respectively.

The quotient space X/G needs not be a manifold, it is then called an orbifold.

If G is a discrete group, the ¬xed points in X under the action of G become

singular points on X/G . For instance, by identifying the points x and ’x of a

plane, the ¬xed point 0 ∈ R2 becomes the tip of the cone R2 /Z2 .

Similar concepts are used for a proper description of the topological space

of the degrees of freedom in gauge theories. Gauge theories contain redundant

variables, i.e. variables which are related to each other by gauge transformations.

This suggests to de¬ne an equivalence relation in the space of gauge ¬elds (cf.

(7) and (90))

Aµ ∼ Aµ if Aµ = A[U ] for some U ,

˜ ˜ (71)

µ

i.e. elements of an equivalence class can be transformed into each other by gauge

transformations U , they are gauge copies of a chosen representative. The equiv-

alence classes

O = A[U ] |U ∈ G (72)

with A ¬xed and U running over the set of gauge transformations are called the

gauge orbits. Their elements describe the same physics. Denoting with A the

space of gauge con¬gurations and with G the space of gauge transformations,

the coset space of gauge orbits is denoted with A/G. It is this space rather than