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u = u0 + ω 0 — x.
2
The helicity density is constant for constant velocity u0 and vorticity ω 0 . For par-
allel velocity and vorticity, the streamlines of the ¬‚uid are right-handed helices.
In magnetohydrodynamics, besides hB and hω , a further topological invariant
the “crossed” helicity can be de¬ned. It characterizes the linkage of ω and B [66].
Finally, I would like to mention the role of the topological charge in the
connection between gauge theories and topological invariants [67,68]. The start-
ing point is the expression (164) for the helicity, which we use as action of the
3-dimensional abelian gauge theory [69], the abelian “Chern“Simons” action

k
d3 x A · B ,
SCS =
8π M

where M is a 3-dimensional manifold and k an integer. One calculates the ex-
pectation value of a product of circular Wilson loops
N
WN = exp i A ds .
Ci
i=1

The Gaussian path integral

D[A]eiSCS WN
WN =
64 F. Lenz

can be performed after inversion of the curl operator (165) in the space of trans-
verse gauge ¬elds. The calculation proceeds along the line of the calculation of
hB (164) and one ¬nds
N
2iπ
∝ exp lk{Ci , Cj } .
WN
k
i=j=1

The path integral for the Chern“Simons theory leads to a representation of a
topological invariant. The key property of the Chern“Simons action is its invari-
ance under general coordinate transformations. SCS is itself a topological invari-
ant. As in other evaluations of expectation values of Wilson loops, determination
of the proportionality constant in the expression for WN requires regulariza-
tion of the path integral due to the linking of each curve with itself (self linking
number). In the extension to non-abelian (3-dimensional) Chern“Simons theory,
the very involved analysis starts with K 0 (150) as the non-abelian Chern“Simons
Lagrangian. The ¬nal result is the Jones“Witten invariant associated with the
product of circular Wilson loops [67].

8 Center Symmetry and Con¬nement
Gauge theories exhibit, as we have seen, a variety of non-perturbative phenom-
ena which are naturally analyzed by topological methods. The common origin
of all the topological excitations which I have discussed is vacuum degeneracy,
i.e. the existence of a continuum or a discrete set of classical ¬elds of minimal
energy. The phenomenon of con¬nement, the trademark of non-abelian gauge
theories, on the other hand, still remains mysterious in spite of large e¬orts
undertaken to con¬rm or disprove the many proposals for its explanation. In
particular, it remains unclear whether con¬nement is related to the vacuum de-
generacy associated with the existence of large gauge transformations or more
generally whether classical or semiclassical arguments are at all appropriate for
its explanation. In the absence of quarks, i.e. of matter in the fundamental
representation, SU (N ) gauge theories exhibit a residual gauge symmetry, the
center symmetry, which is supposed to distinguish between con¬ned and decon-
¬ned phases [70]. Irrespective of the details of the dynamics which give rise to
con¬nement, this symmetry must be realized in the con¬ning phase and sponta-
neously broken in the “plasma” phase. Existence of a residual gauge symmetry
implies certain non-trivial topological properties akin to the non-trivial topo-
logical properties emerging in the incomplete spontaneous breakdown of gauge
symmetries discussed above. In this and the following chapter I will describe
formal considerations and discuss physical consequences related to the center
symmetry properties of SU (2) gauge theory. To properly formulate the center
symmetry and to construct explicitly the corresponding symmetry transforma-
tions and the order parameter associated with the symmetry, the gauge theory
has to be formulated on space-time with (at least) one of the space-time direc-
tions being compact, i.e. one has to study gauge theories at ¬nite temperature
or ¬nite extension.
Topological Concepts in Gauge Theories 65

8.1 Gauge Fields at Finite Temperature and Finite Extension
When heating a system described by a ¬eld theory or enclosing it by making
a spatial direction compact new phenomena occur which to some extent can
be analyzed by topological methods. In relativistic ¬eld theories systems at ¬-
nite temperature and systems at ¬nite extensions with an appropriate choice
of boundary conditions are copies of each other. In order to display the physi-
cal consequences of this equivalence we consider the Stefan“Boltzmann law for
the energy density and pressure for a non-interacting scalar ¬eld with the corre-
sponding quantities appearing in the Casimir e¬ect, i.e. the energy density of the
system if it is enclosed in one spatial direction by walls. I assume the scalar ¬eld
to satisfy periodic boundary conditions on the enclosing walls. The comparison

Stefan“Boltzmann Casimir
π2 4 π 2 ’4
p=’
= T L
15 15
π2 4 π2
= ’ L’4 .
p= T (167)
45 45
expresses a quite general relation between thermal and quantum ¬‚uctuations in
relativistic ¬eld theories [71,72]. This connection is easily established by consid-
ering the partition function given in terms of the Euclidean form (cf. (146)) of
the Lagrangian
β
D[...]e’ dx1 dx2 dx3 LE [...]
dx0
Z= 0

period.
which describes a system of in¬nite extension at temperature T = β ’1 . The
partition function
L
D[...]e’ dx0 dx1 dx2 LE [...]
dx3
Z= 0

period.
describes the same dynamical system in its ground state (T = 0) at ¬nite exten-
sion L in 3-direction. As a consequence, by interchanging the coordinate labels in
the Euclidean, one easily derives allowing for both ¬nite temperature and ¬nite
extension
Z(β, L) = Z(L, β)
(β, L) = ’p(L, β). (168)
These relations hold irrespective of the dynamics of the system. They apply to
non-interacting systems (167) and, more interestingly, they imply that any phase
transition taking place when heating up an interacting system has as counter-
part a phase transition occurring when compressing the system (Quantum phase
transition [41] by variation of the size parameter L). Critical temperature and
critical length are related by
1
Tc = .
Lc
66 F. Lenz

For QCD with its supposed phase transition at about 150 MeV, this relation
predicts the existence of a phase transition when compressing the system beyond
1.3 fm.
Thermodynamic quantities can be calculated as ground state properties of
the same system at the corresponding ¬nite extension. This enables us to apply
the canonical formalism and with it the standard tools of analyzing the system
by symmetry considerations and topological methods. Therefore, in the following
a spatial direction, the 3-direction, is chosen to be compact and of extension L

0 ¤ x3 ¤ L x = (x⊥ , x3 ),

with
x⊥ = (x0 , x1 , x2 ).
Periodic boundary conditions for gauge and bosonic matter ¬elds

Aµ (x⊥ , x3 + L) = Aµ (x⊥ , x3 ) , φ(x⊥ , x3 + L) = φ(x⊥ , x3 ) (169)

are imposed, while fermion ¬elds are subject to antiperiodic boundary conditions

ψ(x⊥ , x3 + L) = ’ψ(x⊥ , x3 ). (170)

In ¬nite temperature ¬eld theory, i.e. for T = 1/L, only this choice of boundary
conditions de¬nes the correct partition functions [73]. The di¬erence in sign of
fermionic and bosonic boundary conditions re¬‚ect the di¬erence in the quanti-
zation of the two ¬elds by commutators and anticommutators respectively. The
negative sign, appearing when going around the compact direction is akin to the
change of sign in a 2π rotation of a spin 1/2 particle.
At ¬nite extension or ¬nite temperature, the ¬elds are de¬ned on S 1 — R3
rather than on R4 if no other compacti¬cation is assumed. Non-trivial topolog-
ical properties therefore emerge in connection with the S 1 component. R3 can
be contracted to a point (cf. (32)) and therefore the cylinder is homotopically
equivalent to a circle
S 1 — Rn ∼ S 1 . (171)
Homotopy properties of ¬elds de¬ned on a cylinder (mappings from S 1 to some
target space) are therefore given by the fundamental group of the target space.
This is illustrated in Fig. 11 which shows two topologically distinct loops. The
loop on the surface of the cylinder can be shrunk to a point, while the loop
winding around the cylinder cannot.

8.2 Residual Gauge Symmetries in QED
I start with a brief discussion of electrodynamics with the gauge ¬elds coupled to
a charged scalar ¬eld as described by the Higgs model Lagrangian (2) (cf. [39,74]).
Due to the homotopic equivalence (171), we can proceed as in our discussion of
1+1 dimensional electrodynamics and classify gauge transformations according
to their winding number and separate the gauge transformations into small and
Topological Concepts in Gauge Theories 67




x⊥




x3
Fig. 11. Polyakov loop (along the compact x3 direction) and Wilson loop (on the
surface of the cylinder) in S 1 — R3


large ones with representative gauge functions given by (158) (with x replaced
by x3 ). If we strictly follow the Faddeev“Popov procedure, gauge ¬xing has to
be carried out by allowing for both type of gauge transformations. Most of the
gauge conditions employed do not lead to such a complete gauge ¬xing. Consider
for instance within the canonical formalism with A0 = 0 the Coulomb-gauge
condition
divA = 0, (172)
and perform a large gauge transformation associated with the representative
gauge function (158)
A(x) ’ A(x) + e3 dn φ(x) ’ eiex3 dn φ(x) . (173)
The transformed gauge ¬eld (cf. (7)) is shifted by a constant and therefore
satis¬es the Coulomb-gauge condition as well. Thus, each gauge orbit O (cf. (72))
is represented by in¬nitely many con¬gurations each one representing a suborbit
On . The suborbits are connected to each other by large gauge transformations,
while elements within a suborbit are connected by small gauge transformations.
The multiple representation of a gauge orbit implies that the Hamiltonian in
Coulomb gauge contains a residual symmetry due to the presence of a residual
redundancy. Indeed, the Hamiltonian in Coulomb gauge containing only the
transverse gauge ¬elds Atr and their conjugate momenta Etr (cf. (10))
12
(E tr + B 2 ) + π — π + (D tr φ)— (D tr φ) + V (φ) ,
H= d3 xH(x) (174)
H=
2
is easily seen to be invariant under the discrete shifts of the gauge ¬elds joined
by discrete rotations of the Higgs ¬eld
[H, eiD3 dn ] = 0. (175)
These transformations are generated by the 3-component of Maxwell™s displace-
ment vector
D = d3 x( E + x j 0 ) ,
68 F. Lenz

with the discrete set of parameters dn given in (158). At this point, the analysis
of the system via symmetry properties is more or less standard and one can
characterize the di¬erent phases of the abelian Higgsmodel by their realization
of the displacement symmetry. It turns out that the presence of the residual
gauge symmetry is necessary to account for the di¬erent phases. It thus appears
that complete gauge ¬xing involving also large gauge transformations is not a
physically viable option.
Like in the symmetry breakdown occurring in the non-abelian Higgs model,
in this procedure of incomplete gauge ¬xing, the U (1) gauge symmetry has not
completely disappeared but the isotropy group Hlgt (69) of the large gauge
transformations (173) generated by D3 remains. Denoting with G1 the (simply
connected) group of gauge transformations in (the covering space) R1 we deduce
from (60) the topological relation
π1 (G1 /Hlgt ) ∼ Z , (176)
which expresses the topological stability of the large gauge transformations.
Equation (176) does not translate directly into a topological stability of gauge
and matter ¬eld con¬gurations. An appropriate Higgs potential is necessary to
force the scalar ¬eld to assume a non-vanishing value. In this case the topo-
logically non-trivial con¬gurations are strings of constant gauge ¬elds winding
around the cylinder with the winding number specifying both the winding of
the phase of the matter ¬eld and the strength of the gauge ¬eld. If, on the
other hand, V (•) has just one minimum at • = 0 nothing prevents a continuous
deformation of a con¬guration to A = • = 0. In such a case, only quantum
¬‚uctuations could possibly induce stability.
Consequences of the symmetry can be investigated without such additional
assumptions. In the Coulomb phase for instance with the Higgs potential given
by the mass term V (φ) = m2 φφ , the periodic potential for the gauge ¬eld
zero-mode
1
a0 = d3 xA3 (x) (177)
3
V
can be evaluated [75]

m2 1
=’ 2 2
Ve¬ (a0 ) cos(neLa0 )K2 (nmL) . (178)
3 3
2
πL n
n=1

The e¬ective potential accounts for the e¬ect of the thermal ¬‚uctuations on the

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( 78 .)



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