boundary condition

H(0) = 0 , K(0) = 1

in the core of the monopole guarantees continuity of the solution. As in the

abelian Higgs model, the changes in the Higgs and gauge ¬eld are occurring

on two di¬erent length scales. Unlike at asymptotic distances, in the core of

the monopole also charged vector ¬elds are present. The core of the monopole

represents the perturbative phase of the Georgi“Glashow model, as the core of

the vortex is made of normal conducting material and ordinary gauge ¬elds.

With the Ansatz (128) the equations of motion are converted into a coupled

system of ordinary di¬erential equations for the unknown functions H and K

which allows for analytical solutions only in certain limits. Such a limiting case

is obtained by saturation of the Bogomol™nyi bound. As for the abelian Higgs

model, this bound is obtained by rewriting the total energy of the static solutions

12 1 1

(B ± Dφ)2 + V (φ) “ BDφ ,

d3 x B + (Dφ)2 + V (φ) = d3 x

E=

2 2 2

and by expressing the last term via an integration by parts (applicable for covari-

ant derivatives due to antisymmetry of the structure constants in the de¬nition

of D in (83)) and with the help of the equation of motion DB = 0 by the

magnetic charge (125)

d3 xB Dφ = a B dσ = a m.

S2

The energy satis¬es the Bogomol™nyi bound

E ≥ |m| a.

For this bound to be saturated, the strength of the Higgs potential has to ap-

proach zero

V = 0, i.e. » = 0,

and the ¬elds have to satisfy the ¬rst order equation

Ba ± (Dφ)a = 0.

In the approach to vanishing », the asymptotics of the Higgs ¬eld |φ| r’∞ a

’’

must remain unchanged. The solution to this system of ¬rst order di¬erential

equations is known as the Prasad“Sommer¬eld monopole

1 sinh agr

H(agr) = coth agr ’ , K(agr) = .

agr agr

50 F. Lenz

In this limiting case of saturation of the Bogomol™nyi bound, only one length scale

exists (ag)’1 . The energy of the excitation, i.e. the mass of the monopole is

given in terms of the mass of the charged vector particles (119) by

4π

E=M .

g2

As for the Nielsen“Olesen vortices, a wealth of further results have been obtained

concerning properties and generalizations of the ™t Hooft“Polyakov monopole

solution. Among them I mention the “Julia“Zee” dyons [43]. These solutions of

the ¬eld equations are obtained using the Ansatz (128) for the Higgs ¬eld and

the spatial components of the gauge ¬eld but admitting a non-vanishing time

component of the form

xa

a

A0 = 2 J(agr).

r

This time component re¬‚ects the presence of a source of electric charge q. Clas-

sically the electric charge of the dyon can assume any value, semiclassical argu-

ments suggest quantization of the charge in units of g [44].

As the vortices of the Abelian Higgs model, ™t Hooft“Polyakov monopoles

induce zero modes if massless fermions are coupled to the gauge and Higgs ¬elds

of the monopole

a

¯„

¯

Lψ = iψγ µ Dµ ψ ’ gφa ψ ψ . (129)

2

The number of zero modes is given by the magnetic charge |m| (125) [45]. Fur-

thermore, the coupled system of a t™ Hooft“Polyakov monopole and a fermionic

zero mode behaves as a boson if the fermions belong to the fundamental represen-

tation of SU (2) (as assumed in (129)) while for isovector fermions the coupled

system behaves as a fermion. Even more puzzling, fermions can be generated

by coupling bosons in the fundamental representation to the ™t Hooft“Polyakov

monopole. The origin of this conversion of isospin into spin [46“48] is the cor-

relation between angular and isospin dependence of Higgs and gauge ¬elds in

solutions of the form (128). Such solutions do not transform covariantly under

spatial rotations generated by J. Under combined spatial and isospin rotations

(generated by I)

K = J + I, (130)

monopoles of the type (128) are invariant. K has to be identi¬ed with the an-

gular momentum operator. If added to this invariant monopole, matter ¬elds

determine by their spin and isospin the angular momentum K of the coupled

system.

Formation of monopoles is not restricted to the particular model. The Georgi“

Glashow model is the simplest model in which this phenomenon occurs. With

the topological arguments at hand, one can easily see the general condition for

the existence of monopoles. If we assume electrodynamics to appear in the pro-

cess of symmetry breakdown from a simply connected topological group G, the

isotropy group H (69) must contain a U (1) factor. According to the identi-

ties (61) and (40), the resulting non trivial second homotopy group of the coset

Topological Concepts in Gauge Theories 51

space

π2 (G/[H — U (1)]) = π1 (H) — Z

˜ ˜ (131)

guarantees the existence of monopoles. This prediction is independent of the

group G, the details of the particular model, or of the process of the symme-

try breakdown. It applies to Grand Uni¬ed Theories in which the structure

of the standard model (SU (3) — SU (2) — U (1)) is assumed to originate from

symmetry breakdown. The fact that monopoles cannot be avoided has posed a

serious problem to the standard model of cosmology. The predicted abundance

of monopoles created in the symmetry breakdown occurring in the early universe

is in striking con¬‚ict with observations. Resolution of this problem is o¬ered by

the in¬‚ationary model of cosmology [49,50].

6 Quantization of Yang“Mills Theory

Gauge Copies. Gauge theories are formulated in terms of redundant variables.

Only in this way, a covariant, local representation of the dynamics of gauge

degrees of freedom is possible. For quantization of the theory both canonically

or in the path integral, redundant variables have to be eliminated. This procedure

is called gauge ¬xing. It is not unique and the implications of a particular choice

are generally not well understood. In the path integral one performs a sum over

all ¬eld con¬gurations. In gauge theories this procedure has to be modi¬ed by

making use of the decomposition of the space of gauge ¬elds into equivalence

classes, the gauge orbits (72). Instead of summing in the path integral over

formally di¬erent but physically equivalent ¬elds, the integration is performed

over the equivalence classes of such ¬elds, i.e. over the corresponding gauge

orbits. The value of the action is gauge invariant, i.e. the same for all members

of a given gauge orbit,

S A[U ] = S [A] .

Therefore, the action is seen to be a functional de¬ned on classes (gauge orbits).

Also the integration measure

d A[U ] = d [A] dAa (x) .

, d [A] = µ

x,µ,a

is gauge invariant since shifts and rotations of an integration variable do not

change the value of an integral. Therefore, in the naive path integral

d [A] eiS[A] ∝

˜

Z= dU (x) .

x

a “volume” associated with the gauge transformations x dU (x) can be fac-

torized and thereby the integration be performed over the gauge orbits. To turn

this property into a working algorithm, redundant variables are eliminated by

imposing a gauge condition

f [A] = 0, (132)

52 F. Lenz

which is supposed to eliminate all gauge copies of a certain ¬eld con¬guration

A. In other words, the functional f has to be chosen such that for arbitrary ¬eld

con¬gurations the equation

f [A [ U ] ] = 0

determines uniquely the gauge transformation U . If successful, the set of all gauge

equivalent ¬elds, the gauge orbit, is represented by exactly one representative.

In order to write down an integral over gauge orbits, we insert into the integral

the gauge-¬xing δ-functional

N 2 ’1

δ [f a (A (x))] .

δ [f (A)] =

x a=1

This modi¬cation of the integral however changes the value depending on the

representative chosen, as the following elementary identity shows

δ (x ’ a)

δ (g (x)) = , g (a) = 0. (133)

|g (a) |

This di¬culty is circumvented with the help of the Faddeev“Popov determinant

∆f [A] de¬ned implicitly by

d [U ] δ f A[U ]

∆f [A] = 1.

˜

Multiplication of the path integral Z with the above “1” and taking into account

the gauge invariance of the various factors yields

˜ d [A] eiS[A] ∆f [A] δ f A[U ]

Z= d [U ]

[U ]

d [A] eiS [A ] ∆f A[U ] δ f A[U ]

= d [U ] = d [U ] Z.

The gauge volume has been factorized and, being independent of the dynamics,

can be dropped. In summary, the ¬nal de¬nition of the generating functional for

gauge theories

d4 xJ µ Aµ

d [A] ∆f [A] δ (f [A] ) eiS[A]+i

Z [J] = (134)

is given in terms of a sum over gauge orbits.

Faddeev“Popov Determinant. For the calculation of ∆f [A], we ¬rst consider

the change of the gauge condition f a [A] under in¬nitesimal gauge transforma-

tions . Taylor expansion

a

δfx [A] b

≈

a [U ] a 4

fx A fx [A] + dy δA (y)

δAb (y) µ

µ

b,µ

a

d4 y M (x, y; a, b) ±b (y)

= fx [A] +

b

Topological Concepts in Gauge Theories 53

with δAa given by in¬nitesimal gauge transformations (92), yields

µ

a

δfx [A]

b,c bcd

Ad

M (x, y; a, b) = ‚µ δ + gf (y) . (135)

µ

δAc (y)

µ

In the second step, we compute the integral

∆’1 [A] = d [U ] δ f A[U ]

f

by expressing the integration d [U ] as an integration over the gauge functions ±.

We ¬nally change to the variables β = M ±

∆’1 [A] = | det M |’1 d [β] δ [f (A) ’ β]

f

and arrive at the ¬nal expression for the Faddeev“Popov determinant

∆f [A] = | det M | . (136)

Examples:

• Lorentz gauge

= ‚ µ Aa (x) ’ χa (x)

a

fx (A) µ

M (x, y; a, b) = ’ δ ab 2 ’ gf abc Ac (y) ‚y δ (4) (x ’ y)

µ

(137)

µ

• Coulomb gauge

= divAa (x) ’ χa (x)

a

fx (A)

M (x, y; a, b) = δ ab ∆ + gf abc Ac (y) ∇y δ (4) (x ’ y) (138)

• Axial gauge

= nµ Aa (x) ’ χa (x)

a

fx (A) µ