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to the relation (222) at least two monopoles with associated strings (cf. (219)).
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

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