of Ve¬ re¬‚ects the residual gauge symmetry. For small amplitude oscillations

eLa0 2π, Ve¬ can be approximated by the quadratic term, which in the

3

small extension or high temperature limit, mL = m/T 1, de¬nes the Debye

screening mass [73,76]

1

m2 = e2 T 2 . (179)

D

3

This result can be obtained by standard perturbation theory. We note that this

perturbative evaluation of Ve¬ violates the periodicity, i.e. it does not respect

the residual gauge symmetry.

Topological Concepts in Gauge Theories 69

8.3 Center Symmetry in SU(2) Yang“Mills Theory

To analyze topological and symmetry properties of gauge ¬xed SU (2) Yang“Mills

theory, we proceed as above, although abelian and non-abelian gauge theories

di¬er in an essential property. Since π1 SU (2) = 0, gauge transformations de-

¬ned on S 1 — R3 are topologically trivial. Nevertheless, non-trivial topological

properties emerge in the course of an incomplete gauge ¬xing enforced by the

presence of a non-trivial center (62) of SU (2). We will see later that this is ac-

tually the correct physical choice for accounting of both the con¬ned and decon-

¬ned phases. Before implementing a gauge condition, it is useful to decompose

the gauge transformations according to their periodicity properties. Although

the gauge ¬elds have been required to be periodic, gauge transformations may

not. Gauge transformations preserve periodicity of gauge ¬elds and of matter

¬elds in the adjoint representation (cf. (89) and (90)) if they are periodic up to

an element of the center of the gauge group

U (x⊥ , L) = cU · U (x⊥ , 0) . (180)

If ¬elds in the fundamental representation are present with their linear depen-

dence on U (89), their boundary conditions require the gauge transformations

U to be strictly periodic cU = 1. In the absence of such ¬elds, gauge transforma-

tions can be classi¬ed according to the value of cU (±1 in SU(2)). An important

example of an SU(2) (cf. (66)) gauge transformation u’ with c = ’1 is

ˆ

u’ = eiπψ„ x3 /L = cos πx3 /L + iψ„ sin πx3 /L.

ˆ (181)

Here ψ(x⊥ ) is a unit vector in color space. For constant ψ, it is easy to verify

ˆ ˆ

that the transformed gauge ¬elds

πˆ

ˆ ˆ

Aµ ’ ] = eiπψ„ x3 /L Aµ e’iπψ„ x3 /L ’ ψ„ δµ3

[u

gL

indeed remain periodic and continuous. Locally, cU = ±1 gauge transformations

U cannot be distinguished. Global changes induced by gauge transformations

like (181) are detected by loop variables winding around the compact x3 direc-

tion. The Polyakov loop,

L

P (x⊥ ) = P exp ig dx3 A3 (x) , (182)

0

is the simplest of such variables and of importance in ¬nite temperature ¬eld

theory. The coordinate x⊥ denotes the position of the Polyakov loop in the space

transverse to x3 . Under gauge transformations (cf. (94) and (96))

P (x⊥ ) ’ U (x⊥ , L) P (x⊥ )U † (x⊥ , 0) .

With x = (x⊥ , 0) and x = (x⊥ , L) labeling identical points, the Polyakov loop is

seen to distinguish cU = ±1 gauge transformations. In particular, we have

SU(2)

tr{P (x⊥ )} ’ tr{cU P (x⊥ )} = ±tr{P (x⊥ )}.

70 F. Lenz

With this result, we now can transfer the classi¬cation of gauge transformations

to a classi¬cation of gauge ¬elds. In SU (2), the gauge orbits O (cf. (72)) are

decomposed according to c = ±1 into suborbits O± . Thus these suborbits are

characterized by the sign of the Polyakov loop at some ¬xed reference point x0⊥

A(x) ∈ O± ± tr{P (x0 )} ≥ 0.

, if (183)

⊥

Strictly speaking, it is not the trace of the Polyakov loop rather only its modulus

|tr{P (x⊥ )}| which is invariant under all gauge transformations. Complete gauge

¬xing, i.e. a representation of gauge orbits O by exactly one representative, is

only possible if the gauge ¬xing transformations are not strictly periodic. In turn,

if gauge ¬xing is carried out with strictly periodic gauge ¬xing transformations

(U, cU = 1) the resulting ensemble of gauge ¬elds contains one representative

Af for each of the suborbits (183). The label f marks the dependence of the

±

representative on the gauge condition (132). The (large) cU = ’1 gauge trans-

formation mapping the representatives of two gauge equivalent suborbits onto

each other are called center re¬‚ections

Z : Af ” Af . (184)

’

+

Under center re¬‚ections

tr P (x⊥ ) ’ ’tr P (x⊥ ).

Z: (185)

The center symmetry is a standard symmetry within the canonical formalism.

Center re¬‚ections commute with the Hamiltonian

[H, Z] = 0 . (186)

Stationary states in SU(2) Yang“Mills theory can therefore be classi¬ed accord-

ing to their Z-Parity

H|n± = En± |n± , Z|n± = ±|n± . (187)

The dynamics of the Polyakov loop is intimately connected to con¬nement.

The Polyakov loop is associated with the free energy of a single heavy charge. In

electrodynamics, the coupling of a heavy pointlike charge to an electromagnetic

¬eld is given by

L

d xδ(x ’ y) A0 (x) = e

4 µ 4

δL = d xj Aµ = e dx0 A0 (x0 , y) ,

0

which, in the Euclidean and after interchange of coordinate labels 0 and 3,

reduces to the logarithm of the Polyakov loop. The property of the system to

con¬ne can be formulated as a symmetry property. The expected in¬nite free

energy of a static color charge results in a vanishing ground state expectation

value of the Polyakov loop

0|tr P (x⊥ )|0 = 0 (188)

Topological Concepts in Gauge Theories 71

in the con¬ned phase. This property is guaranteed if the vacuum is center sym-

metric. The interaction energy V (x⊥ ) of two static charges separated in a trans-

verse direction is, up to an additive constant, given by the Polyakov-loop corre-

lator

0|trP (x⊥ )trP (0)|0 = e’LV (x⊥ ) . (189)

Thus, vanishing of the Polyakov-loop expectation values in the center symmetric

phase indicates an in¬nite free energy of static color charges, i.e. con¬nement.

For non-zero Polyakov-loop expectation values, the free energy of a static color

charge is ¬nite and the system is decon¬ned. A non-vanishing expectation value

is possible only if the center symmetry is broken. Thus, in the transition from

the con¬ned to the plasma phase, the center symmetry, i.e. a discrete part of the

underlying gauge symmetry, must be spontaneously broken. As in the abelian

case, a complete gauge ¬xing, i.e. a de¬nition of gauge orbits including large

gauge transformations may not be desirable or even possible. It will prevent a

characterization of di¬erent phases by their symmetry properties.

As in QED, non-trivial residual gauge symmetry transformations do not nec-

essarily give rise to topologically non-trivial gauge ¬elds. For instance, the pure

gauge obtained from the non-trivial gauge transformation (181), with constant

ψ, Aµ = ’ gL ψ„ δµ3 is deformed trivially, along a path of vanishing action, into

ˆ πˆ

Aµ = 0. In this deformation, the value of the Polyakov loop (182) changes con-

tinuously from ’1 to 1. Thus a vacuum degeneracy exists with the value of the

Polyakov loop labeling the gauge ¬elds of vanishing action. A mechanism, like

the Higgs mechanism, which gives rise to the topological stability of excitations

built upon the degenerate classical vacuum has not been identi¬ed.

8.4 Center Vortices

Here, we again view the (incomplete) gauge ¬xing process as a symmetry break-

down which is induced by the elimination of redundant variables. If we require

the center symmetry to be present after gauge ¬xing, the isotropy group formed

by the center re¬‚ections must survive the “symmetry breakdown”. In this way,

we e¬ectively change the gauge group

SU (2) ’ SU (2)/Z(2). (190)

Since π1 SU (2)/Z2 = Z2 , as we have seen (63), this space of gauge transfor-

mations contains topologically stable defects, line singularities in R3 or singular

sheets in R4 . Associated with such a singular gauge transformation UZ2 (x) are

pure gauges (with the singular line or sheet removed)

1 †

Aµ2 (x) = UZ2 (x) ‚ µ UZ2 (x).

Z

ig

The following gauge transformation written in cylindrical coordinates ρ, •, z, t

•3

UZ2 (•) = exp i „

2

72 F. Lenz

exhibits the essential properties of singular gauge transformations, the center

vortices, and their associated singular gauge ¬elds . UZ2 is singular on the sheet

ρ = 0 ( for all z, t). It has the property

UZ2 (2π) = ’UZ2 (0),

i.e. the gauge transformation is continuous in SU (2)/Z2 but discontinuous as

an element of SU (2). The Wilson loop detects the defect. According to (97)

and (98), the Wilson loop, for an arbitrary path C enclosing the vortex, is given

by

1 †

WC, Z2 = tr UZ2 (2π) UZ2 (0) = ’1 . (191)

2

The corresponding pure gauge ¬eld has only one non-vanishing space-time com-

ponent

13

A•2 (x) = ’ „, (192)

Z

2gρ

which displays the singularity. For calculation of the ¬eld strength, we can, with

only one color component non-vanishing, apply Stokes theorem. We obtain for

the ¬‚ux through an area of arbitrary size Σ located in the x ’ y plane

π

F12 ρdρd• = ’ „ 3 ,

g

Σ

and conclude

π

F12 = ’ „ 3 δ (2) (x).

g

This divergence in the ¬eld strength makes these ¬elds irrelevant in the sum-

mation over all con¬gurations. However, minor changes, like replacing the 1/ρ

in A•2 by a function interpolating between a constant at ρ = 0 and 1/ρ at

Z

large ρ eliminate this singularity. The modi¬ed gauge ¬eld is no longer a pure

gauge. Furthermore, a divergence in the action from the in¬nite extension can

be avoided by forming closed ¬nite sheets. All these modi¬cations can be carried

out without destroying the property (191) that the Wilson loop is ’1 if enclos-

ing the vortex. This crucial property together with the assumption of a random

distribution of center vortices yields an area law for the Wilson loop. This can

be seen (cf. [77]) by considering a large area A in a certain plane containing a

loop of much smaller area AW . Given a ¬xed number N of intersection points

of vortices with A, the number of intersection points with AW will ¬‚uctuate

and therefore the value W of the Wilson loop. For a random distribution of

intersection points, the probability to ¬nd n intersection points in AW is given

by

AW n AW N ’n

N

1’

pn = .

A A

n

Since, as we have seen, each intersection point contributes a factor ’1, one

obtains in the limit of in¬nite A with the density ν of intersection points, i.e.

Topological Concepts in Gauge Theories 73

vortices per area kept ¬xed,

N

(’1)n pn ’ exp ’ 2νAW .

W=

n=1

As exempli¬ed by this simple model, center vortices, if su¬ciently abundant and

su¬ciently disordered, could be responsible for con¬nement (cf. [78]).

It should be noticed that, unlike the gauge transformation UZ2 , the associated

pure gauge Aµ2 is not topologically stable. It can be deformed into Aµ = 0 by a

Z

continuous change of its strength. This deformation, changing the magnetic ¬‚ux,

is not a gauge transformation and therefore the stability of UZ2 is compatible

with the instability of AZ2 . In comparison to nematic substances with their stable

Z2 defects (cf. Fig. 7), the degrees of freedom of Yang“Mills theories are elements

of the Lie algebra and not group-elements and it is not unlikely that the stability

of Z2 vortices pertains only to formulations of Yang“Mills theories like lattice

gauge theories where the elementary degrees of freedom are group elements.

It is instructive to compare this unstable defect in the gauge ¬eld with a

topologically stable vortex. In a simple generalization [8] of the non-abelian

Higgs model (82) such vortices appear. One considers a system containing two

instead of one Higgs ¬eld with self-interactions of the type (104)

1 »k 2

Lm = Dµ φk Dµ φk ’ (φk ’ a2 )2 ’ V12 (φ1 φ2 ) , »k > 0 . (193)

k

2 4

k=1,2

By a choice of the interaction between the two scalar ¬elds which favors the

Higgs ¬elds to be orthogonal to each other in color space, a complete spontaneous

symmetry breakdown up to multiplication of the Higgs ¬elds with elements of

the center of SU (2) can be achieved. The static, cylindrically symmetric Ansatz

for such a “Z2 -vortex” solution [79]

a1 3 a2 1

A• = ’

f (ρ) cos • „ 1 + sin • „ 2 , ±(ρ)„ 3

φ1 = „, φ2 = (194)

2 2 2g