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gauge ¬eld zero-mode. It vanishes at zero temperature (L ’ ∞). The periodicity
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
small extension or high temperature limit, mL = m/T 1, de¬nes the Debye
screening mass [73,76]
m2 = e2 T 2 . (179)
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
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
ˆ ˆ
Aµ ’ ] = eiπψ„ x3 /L Aµ e’iπψ„ x3 /L ’ ψ„ δµ3
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,
P (x⊥ ) = P exp ig dx3 A3 (x) , (182)

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
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
d xδ(x ’ y) A0 (x) = e
4 µ 4
δL = d xj Aµ = e dx0 A0 (x0 , y) ,

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-
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).
The following gauge transformation written in cylindrical coordinates ρ, •, z, t
UZ2 (•) = exp i „
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
1 †
WC, Z2 = tr UZ2 (2π) UZ2 (0) = ’1 . (191)
The corresponding pure gauge ¬eld has only one non-vanishing space-time com-
A•2 (x) = ’ „, (192)
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

F12 ρdρd• = ’ „ 3 ,

and conclude
F12 = ’ „ 3 δ (2) (x).
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
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
AW n AW N ’n
pn = .
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,
(’1)n pn ’ exp ’ 2νAW .

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
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)
2 4

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


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