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