would decouple

π

∆E ≈ ’ ∞.

L

Only neutral gluon ¬elds are periodic in the compact x3 direction and therefore

possess zero modes. Thus, at small extension or high temperature L ’ 0, only

neutral gluons would contribute to thermodynamic quantities. This is in con-

¬‚ict with results of lattice gauge calculations [90] and we therefore will assume

that the center symmetry is spontaneously broken for L ¤ Lc = 1/Tc . In the

high-temperature phase, Aharonov“Bohm ¬‚uxes must be screened and the geo-

metrical mass must be reduced. Furthermore, with the string tension vanishing in

the plasma phase, the e¬ects of the Haar measure must be e¬ectively suppressed

and the Polyakov-loop variables may be treated as classical ¬elds. On the basis

84 F. Lenz

of this assumption, I now describe the development of a phenomenological treat-

ment of the plasma phase [91]. For technical simplicity, I will neglect the space

time dependence of a3 and describe the results for vanishing geometrical mass

M . For the description of the high-temperature phase it is more appropriate to

use the variables

χ = gLa3 + π

with vanishing average Aharonov“Bohm ¬‚ux. Charged gluons satisfy quasi-pe-

riodic boundary conditions

A1,2 (x⊥ , x3 + L) = eiχ A1,2 (x⊥ , x3 ). (216)

µ µ

Furthermore, we will calculate the thermodynamic properties by evaluation of

the energy density in the Casimir e¬ect (cf. (167) and (168)). In the Casimir

e¬ect, the central quantity to be calculated is the ground state energy of gluons

between plates on which the ¬elds have to satisfy appropriate boundary condi-

tions. In accordance with our choice of boundary conditions (169), we assume

the enclosing plates to extend in the x1 and x2 directions and to be separated

in the x3 direction. The essential observation for the following phenomenological

description is the dependence of the Casimir energy on the boundary conditions

and therefore on the presence of Aharonov“Bohm ¬‚uxes. The Casimir energy of

the charged gluons is obtained by summing, after regularization, the zero point

energies

∞ 1/2

d2 k⊥ k2 + (2πn + χ)2 4π 2

1 χ

⊥

µ(L, χ) = = B4 (217)

2 L2 3L4

2 n=’∞ (2π) 2π

with

1

B4 (x) = ’ + x2 (1 ’ x)2 .

30

Thermodynamic stability requires positive pressure at ¬nite temperature and

thus, according to (168), a negative value for the Casimir energy density. This

requirement is satis¬ed if

χ ¤ 1.51.

For complete screening ( χ = 0 ) of the Aharonov“Bohm ¬‚uxes, the expression for

the pressure in black-body radiation is obtained (the factor of two di¬erence be-

tween (167) and (217) accounts for the two charged gluonic states). Unlike QED,

QCD is not stable for vanishing Aharonov“Bohm ¬‚uxes. In QCD the perturba-

tive ground state energy can be lowered by spontaneous formation of magnetic

¬elds. Magnetic stability can be reached if the strength of the Aharonov“Bohm

¬‚uxes does not decrease beyond a certain minimal value. By calculating the

Casimir e¬ect in the presence of an external, homogeneous color-magnetic ¬eld

Bi = δ a3 δi3 B ,

a

this minimal value can be determined. The energy of a single quantum state

is given in terms of the oscillator quantum number m for the Landau orbits,

Topological Concepts in Gauge Theories 85

Fig. 14. Left: Regions of stability and instability in the (L, χ) plane. To the right of

the circles, thermodynamic instability; above the solid line, magnetic instability. Right:

Energy density and pressure normalized to Stefan“Boltzmann values vs. temperature

in units of ΛMS

in terms of the momentum quantum number n for the motion in the (compact)

direction of the magnetic ¬eld, and by a magnetic moment contribution (s = ±1)

1/2

(2πn + χ)2

Emns = 2gH(m + 1/2) + + 2sgH .

L2

This expression shows that the destabilizing magnetic moment contribution

2sgH in the state with

s = ’1, m = 0, n = 0

can be compensated by a non-vanishing Aharonov“Bohm ¬‚ux χ of su¬cient

strength. For determination of the actual value of χ, the sum over these energies

has to be performed. After regularizing the expression, the Casimir energy den-

sity can be computed numerically. The requirement of magnetic stability yields

a lower limit on χ. As Fig. 14 shows, the Stefan“Boltzmann limit χ = 0 is not

compatible with magnetic stability for any value of the temperature. Identi¬-

cation of the Aharonov“Bohm ¬‚ux with the minimal allowed values sets upper

limits to energy density and pressure which are shown in Fig. 14. These results

are reminiscent of lattice data [92] in the slow logarithmic approach of energy

density and pressure

11 2

χ(T ) ≥ g (T ), T ’ ∞

12

to the Stefan“Boltzmann limit.

It appears that the ¬nite value of the Aharonov“Bohm ¬‚ux accounts for in-

teractions present in the decon¬ned phase fairly well; qualitative agreement with

86 F. Lenz

lattice calculations is also obtained for the “interaction measure” ’ 3P . Fur-

thermore, these limits on χ also yield a realistic estimate for the change in energy

density ’∆ across the phase transition. The phase transition is accompanied

by a change in strength of the Aharonov“Bohm ¬‚ux from the center symmetric

value π to a value in the stability region. The lower bound is determined by

thermodynamic stability

7π 2 1

’∆ ≥ (Lc , χ = π) ’ (Lc , χ = 1.51) = .

180 L4c

For establishing an upper bound, the critical temperature must be speci¬ed. For

Tc ≈ 270 MeV,

1 1

0.38 4 ¤ ’∆ ¤ 0.53 4 .

Lc Lc

These limits are compatible with the lattice result [93]

1

∆ = ’0.45 .

L4

c

The picture of increasing Debye screening of the Aharonov“Bohm ¬‚uxes with in-

creasing temperature seems to catch the essential physics of the thermodynamic

quantities. It is remarkable that the requirement of magnetic stability, which

prohibits complete screening, seems to determine the temperature dependence

of the Aharonov“Bohm ¬‚uxes and thereby to simulate the non-perturbative dy-

namics in a semiquantitative way.

9.4 Monopoles

The discussion of the dynamics of the Polyakov loops has demonstrated that

signi¬cant changes occur if compact variables are present. The results discussed

strongly suggest that con¬nement arises naturally in a setting where the dy-

namics is dominated by such compact variables. The Polyakov-loop variables

a3 (x⊥ ) constitute only a small set of degrees of freedom in gauge theories. In

axial gauge, the remaining degrees of freedom A⊥ (x) are standard ¬elds which,

with interactions neglected, describe freely propagating particles. As a conse-

quence, the coupling of the compact variables to the other degrees of freedom

almost destroys the con¬nement present in the system of uncoupled Polyakov-

loop variables. This can be prevented to happen only if mechanisms are opera-

tive by which all the gluon ¬elds acquire infrared properties similar to those of

the Polyakov-loop variables. In the axial gauge representation it is tempting to

connect such mechanisms to the presence of monopoles whose existence is inti-

mately linked to the compactness of the Polyakov-loop variables. In analogy to

the abelian Higgs model, condensation of magnetic monopoles could be be a ¬rst

and crucial element of a mechanism for con¬nement. It would correspond to the

formation of the charged Higgs condensate |φ| = a (13) enforced by the Higgs

self-interaction (3). Furthermore, this magnetically charged medium should dis-

play excitations which behave as chromo-electric vortices. Concentration of the

Topological Concepts in Gauge Theories 87

electric ¬eld lines to these vortices ¬nally would give rise to a linear increase in

the interaction energy of two chromo-electric charges with their separation as in

(31). These phenomena actually happen in the Seiberg“Witten theory [94]. The

Seiberg“Witten theory is a supersymmetric generalization of the non-abelian

Higgs model. Besides gauge and Higgs ¬elds it contains fermions in the adjoint

representation. It exhibits vacuum degeneracy enlarged by supersymmetry and

contains topologically non-trivial excitations, both monopoles and instantons.

The monopoles can become massless and when partially breaking the supersym-

metry, condensation of monopoles occurs that induces con¬nement of the gauge

degrees of freedom.

In this section I will sketch the emergence of monopoles in axial gauge and

discuss some elements of their dynamics. Singular ¬eld arise in the last step of the

gauge ¬xing procedure (200), where the variables characterizing the orientation

of the Polyakov loops in color space are eliminated as redundant variables by

diagonalization of the Polyakov loops. The diagonalization of group elements is

achieved by the unitary matrix

„¦D = eiω„ = cos ω + i„ ω sin ω ,

ˆ

with ω(x⊥ ) depending on the Polyakov loop P (x⊥ ) to be diagonalized. This

diagonalization is ill de¬ned if

P (x⊥ ) = ±1 , (218)

i.e. if the Polyakov loop is an element of the center of the group (cf. (62)).

Diagonalization of an element in the neighborhood of the center of the group is

akin to the de¬nition of the azimuthal angle on the sphere close to the north or

south pole. With „¦D ill de¬ned, the transformed ¬elds

1 †

Aµ (x) = „¦D (x⊥ ) Aµ (x) + ‚µ „¦D (x⊥ )

ig

develop singularities. The most singular piece arises from the inhomogeneous

term in the gauge transformation

1 †

sµ (x⊥ ) = „¦D (x⊥ ) ‚µ „¦D (x⊥ ) .

ig

For a given a3 (x⊥ ) with orientation described by polar θ(x⊥ ) and azimuthal

angles •(x⊥ ) in color space, the matrix diagonalizing a3 (x⊥ ) can be represented

as

ei• cos(θ/2) sin(θ/2)

„¦D =

’ sin(θ/2) e’i• cos(θ/2)

and therefore the nature of the singularities can be investigated in detail. The

condition for the Polyakov loop to be in the center of the group, i.e. at a de¬nite

point on S 3 (218), determines in general uniquely the corresponding position in

R3 and therefore the singularities form world-lines in 4-dimensional space-time.

88 F. Lenz

The singularities are “monopoles” with topologically quantized charges. „¦D is

determined only up to a gauge transformation

3

„¦D (x⊥ ) ’ ei„ ψ(x⊥ )

„¦D (x⊥ )

and is therefore an element of SU (2)/U (1). The mapping of a sphere S 2 around

the monopole in x⊥ space to SU (2)/U (1) is topologically non-trivial

π2 [SU (2)/U (1)] = Z (67). This argument is familiar to us from the discussion

of the ™t Hooft“Polyakov monopole (cf. (125) and (126)). Also here we identify

the winding number associated with this mapping as the magnetic charge of the

monopole.

Properties of Singular Fields

• Dirac monopoles, extended to include color, constitute the simplest examples

of singular ¬elds (Euclidean x⊥ = x)

m 1 + cos θ

•„ 3 + [(• + iθ)e’i• („ 1 ’ i„ 2 ) + h.c.] .

ˆ

A∼ ˆ ˆ (219)

2gr sin θ

In addition to the pole at r = 0, the ¬elds contain a Dirac string in 3-space

(here chosen along θ = 0) and therefore a sheet-like singularity in 4-space

which emanates from the monopole word-line.

• Monopoles are characterized by two charges, the “north-south” charge for

the two center elements of SU(2) (218),

z = ±1 , (220)

and the quantized strength of the singularity

m = ±1, ±2, .... . (221)

• The topological charge (149) is determined by the two charges of the mono-

poles present in a given con¬guration [95“97]

1

ν= mi z i . (222)

2 i

Thus, after elimination of the redundant variables, the topological charge

resides exclusively in singular ¬eld con¬gurations.

• The action of singular ¬elds is in general ¬nite and can be arbitrarily small

for ν = 0. The singularities in the abelian and non-abelian contributions

to the ¬eld strength cancel since by gauge transformations singularities in

gauge covariant quantities cannot be generated.

Topological Concepts in Gauge Theories 89

9.5 Monopoles and Instantons

By the gauge choice, i.e. by the diagonalization of the Polyakov loop by „¦D

in (200), monopoles appear; instantons, which in (singular) Lorentz gauge have

a point singularity (154) at the center of the instanton, must possess according