Thus, in axial gauge, an instanton ¬eld becomes singular on world lines and world

sheets. To illustrate the connection between topological charges and monopole

charges (222), we consider the singularity content of instantons in axial gauge [64]

and calculate the Polyakov loop of instantons. To this end, the generalization of

the instantons (154) to ¬nite temperature (or extension) is needed. The so-called

“calorons” are known explicitly [98]

1 (sinh u)/u

·µν ∇ν ln 1 + γ

Aµ = ¯ (223)

cosh u ’ cos v

g

where

u = 2π|x⊥ ’ x0 |/L , γ = 2(πρ/L)2 .

v = 2πx3 /L ,

⊥

The topological charge and the action are independent of the extension,

8π 2

ν = 1, S= 2 .

g

The Polyakov loops can be evaluated in closed form

x⊥ ’ x0 „

⊥

P (x) = exp iπ χ(u) , (224)

|x⊥ ’ x0 |

⊥

with

(1 ’ γ/u2 ) sinh u + γ/u cosh(u)

χ(u) = 1 ’ .

(cosh u + γ/u sinh u)2 ’ 1

As Fig. 15 illustrates, instantons contain a z = ’1 monopole at the center and

a z = 1 monopole at in¬nity; these monopoles carry the topological charge of the

instanton. Furthermore, tunneling processes represented by instantons connect

¬eld con¬gurations of di¬erent winding number (cf. (151)) but with the same

value for the Polyakov loop. In the course of the tunneling, the Polyakov loop

of the instanton may pass through or get close to the center element z = ’1, it

however always returns to its original value z = +1. Thus, instanton ensembles

in the dilute gas limit are not center symmetric and therefore cannot give rise to

con¬nement. One cannot rule out that the z = ’1 values of the Polyakov loop

are encountered more and more frequently with increasing instanton density.

In this way, a center-symmetric ensemble may ¬nally be reached in the high-

density limit. This however seems to require a ¬ne tuning of instanton size and

the average distance between instantons.

90 F. Lenz

0.5

0

-0.5

-1

-4 -2 0 2 4

Fig. 15. Polyakov loop (224) of an instanton (223) of “size” γ = 1 as a function of

time t = 2πx0 /L for minimal distance to the center 2πx1 /L = 0 (solid line), L = 1

(dashed line), L = 2 (dotted line), x2 = 0

9.6 Elements of Monopole Dynamics

In axial gauge, instantons are composed of two monopoles. An instanton gas

(163) of ¬nite density nI therefore contains ¬eld con¬gurations with in¬nitely

many monopoles. The instanton density in 4-space can be converted approxi-

mately to a monopole density in 3-space [97]

3/2

nM ∼ (LnI ρ) , ρ L,

nM ∼ LnI ρ ≥ L.

,

With increasing extension or equivalently decreasing temperature, the monopole

density diverges for constant instanton density. Nevertheless, the action density

of an instanton gas remains ¬nite. This is in accordance with our expectation

that production of monopoles is not necessarily suppressed by large values of the

action. Furthermore, a ¬nite or possibly even divergent density of monopoles as

in the case of the dilute instanton gas does not imply con¬nement.

Beyond the generation of monopoles via instantons, the system has the addi-

tional option of producing one type (z = +1 or z = ’1) of poles and correspond-

ing antipoles only. No topological charge is associated with such singular ¬elds

and their occurrence is not limited by the instanton bound ((147) and (152))

on the action as is the case for a pair of monopoles of opposite z-charge. Thus,

entropy favors the production of such con¬gurations. The entropy argument also

applies in the plasma phase. For purely kinematical reasons, a decrease in the

monopole density must be expected as the above estimates within the instanton

model show. This decrease is counteracted by the enhanced probability to pro-

duce monopoles when, with decreasing L, the Polyakov loop approaches more

and more the center of the group, as has been discussed above (cf. left part of

Fig. 14). A ¬nite density of singular ¬elds is likely to be present also in the de-

con¬ned phase. In order for this to be compatible with the partially perturbative

nature of the plasma phase and with dimensional reduction to QCD2+1 , poles

Topological Concepts in Gauge Theories 91

and antipoles may have to be strongly correlated with each other and to form

e¬ectively a gas of dipoles.

Since entropy favors proliferate production of monopoles and monopoles may

be produced with only a small increase in the total action, the coupling of the

monopoles to the quantum ¬‚uctuations must ultimately prevent unlimited in-

crease in the number of monopoles. A systematic study of the relevant dynamics

has not been carried out. Monopoles are not solutions to classical ¬eld equations.

Therefore, singular ¬elds are mixed with quantum ¬‚uctuations even on the level

of bilinear terms in the action. Nevertheless, two mechanisms can be identi¬ed

which might limit the production of monopoles.

• The 4-gluon vertex couples pairs of monopoles to charged and neutral gluons

and can generate masses for all the color components of the gauge ¬elds. A

simple estimate yields

N

π

δm = ’ mi mj |x⊥i ’ x⊥j |

2

V i,j=1

i<j

with the monopole charges mi and positions x⊥ i . If operative also in the

decon¬ned phase, this mechanism would give rise to a magnetic gluon mass.

• In general, ¬‚uctuations around singular ¬elds generate an in¬nite action.

Finite values of the action result only if the ¬‚uctuations δφ, δA3 satisfy the

boundary conditions,

δφ(x) e2i•(x⊥ ) continuous along the strings ,

= δA3

δφ(x) = 0.

at pole at pole

For a ¬nite monopole density, long wave-length ¬‚uctuations cannot simulta-

neously satisfy boundary conditions related to monopoles or strings which

are close to each other. One therefore might suspect quantum ¬‚uctuations

with wavelengths

’1/3

» ≥ »max = nM

to be suppressed.

We note that both mechanisms would also suppress the propagators of the quan-

tum ¬‚uctuations in the infrared. Thereby, the decrease in the string constant by

coupling Polyakov loops to charged gluons could be alleviated if not cured.

9.7 Monopoles in Diagonalization Gauges

In axial gauge, monopoles appear in the gauge ¬xing procedure (200) as defects

in the diagonalization of the Polyakov loops. Although the choice was motivated

by the distinguished role of the Polyakov-loop variables as order parameters,

92 F. Lenz

formally one may choose any quantity φ which, if local, transforms under gauge

transformations U as

φ ’ U φU † ,

where φ could be either an element of the algebra or of the group. In analogy

to (202), the gauge condition can be written as

„3

f [φ] = φ ’ • , (225)

2

with arbitrary • to be integrated in the generating functional. A simple illustra-

tive example is [99]

φ = F12 . (226)

The analysis of the defects and the resulting properties of the monopoles can

be taken over with minor modi¬cations from the procedure described above.

Defects occur if

φ=0

(or φ = ±1 for group elements). The condition for the defect is gauge invari-

ant. Generically, the three defect conditions determine for a given gauge ¬eld

the world-lines of the monopoles generated by the gauge condition (225). The

quantization of the monopole charge is once more derived from the topological

identity (67) which characterizes the mapping of a (small) sphere in the space

transverse to the monopole world-line and enclosing the defect. The coset space,

appears as above since the gauge condition leaves a U (1) gauge symmetry re-

lated to the rotations around the direction of φ unspeci¬ed. With φ being an

element of the Lie algebra, only one sort of monopoles appears. The character-

ization as z = ±1 monopoles requires φ to be an element of the group. As a

consequence, the generalization of the connection between monopoles and topo-

logical charges is not straightforward. It has been established [20] with the help

of the Hopf-invariant (cf. (45)) and its generalization.

It will not have escaped the attention of the reader that the description of

Yang“Mills theories in diagonalization gauges is almost in one to one correspon-

dence to the description of the non-abelian Higgs model in the unitary gauge.

In particular, the gauge condition (225) is essentially identical to the unitary

gauge condition (115). However, the physics content of these gauge choices is

very di¬erent. The unitary gauge is appropriate if the Higgs potential forces

the Higgs ¬eld to assume (classically) a value di¬erent from zero. In the clas-

sical limit, no monopoles related to the vanishing of the Higgs ¬eld appear in

unitary gauge and one might expect that quantum ¬‚uctuations will not change

this qualitatively. Associated with the unspeci¬ed U (1) are the photons in the

Georgi“Glashow model. In pure Yang“Mills theory, gauge conditions like (226)

are totally inappropriate in the classical limit, where vanishing action produces

defects ¬lling the whole space. Therefore, in such gauges a physically meaning-

ful condensate of magnetic monopoles signaling con¬nement can arise only if

quantum ¬‚uctuations change the situation radically. Furthermore, the unspec-

i¬ed U (1) does not indicate the presence of massless vector particles, it rather

Topological Concepts in Gauge Theories 93

re¬‚ects an incomplete gauge ¬xing. Other diagonalization gauges may be less

singular in the classical limit, like the axial gauge. However, independent of the

gauge choice, defects in the gauge condition have not been related convincingly

to physical properties of the system. They exist as as coordinate singularities

and their physical signi¬cance remains enigmatic.

10 Conclusions

In these lecture notes I have described the instanton, the ™t Hooft“Polyakov

monopole, and the Nielsen“Olesen vortex which are the three paradigms of topo-

logical objects appearing in gauge theories. They di¬er from each other in the

dimensionality of the core of these objects, i.e. in the dimension of the subman-

ifold of space-time on which gauge and/or matter ¬elds are singular. This di-

mension is determined by the topological properties of the spaces in which these

¬elds take their values and dictates to a large extent the dynamical role these

objects can play. ™t Hooft“Polyakov monopoles are singular along a world-line

and therefore describe particles. I have presented the strong theoretical evidence

based on topological arguments that these particles have been produced most

likely in phase transitions of the early universe. These relics of the big bang

have not been and most likely cannot be observed. Their abundance has been

diluted in the in¬‚ationary phase. Nielsen“Olesen vortices are singular on lines in

space or equivalently on world-sheets in space-time. Under suitable conditions

such objects occur in Type II superconductors. They give rise to various phases

and a wealth of phenomena in superconducting materials. Instantons become

singular on a point in Euclidean 4-space and they therefore represent tunneling

processes. In comparison to monopoles and vortices, the manifestation of these

objects is only indirect. They cannot be observed but are supposed to give rise

to non-perturbative properties of the corresponding quantum mechanical ground

state.

Despite their di¬erence in dimensionality, these topological objects have

many properties in common. They are all solutions of the non-linear ¬eld equa-

tions of gauge theories. They owe their existence and topological stability to

vacuum degeneracy, i.e. the presence of a continuous or discrete set of distinct

solutions with minimal energy. They can be classi¬ed according to a charge,

which is quantized as a consequence of the non-trivial topology. Their non-trivial

properties leave a topological imprint on fermionic or bosonic degrees of free-

dom when coupled to these objects. Among the topological excitations of a given

type, a certain class is singled out by their energy determined by the quantized

charge.

In these lecture notes I also have described e¬orts in the topological analysis

of QCD. A complete picture about the role of topologically non-trivial ¬eld con-

¬gurations has not yet emerged from such studies. With regard to the breakdown

of chiral symmetry, the formation of quark condensates and other chiral prop-

erties, these e¬orts have met with success. The relation between the topological

charge and fermionic properties appears to be at the origin of these phenom-

94 F. Lenz

ena. The instanton model incorporates this connection explicitly by reducing

the quark and gluon degrees of freedom to instantons and quark zero modes

generated by the topological charge of the instantons. However, a generally ac-

cepted topological explanation of con¬nement has not been achieved nor have

¬eld con¬gurations been identi¬ed which are relevant for con¬nement. The neg-

ative outcome of such investigations may imply that, unlike mass generation

by the Higgs mechanism, con¬nement does not have an explanation within the

context of classical ¬eld theory. Such a conclusion is supported by the simple