M (x, y; a, b) = ’δ ab nµ ‚y δ (4) (x ’ y)

µ

(139)

We note that in axial gauge, the Faddeev“Popov determinant does not de-

pend on the gauge ¬elds and therefore changes the generating functional only

by an irrelevant factor.

Gribov Horizons. As the elementary example (133) shows, a vanishing

Faddeev“Popov determinant g (a) = 0 indicates the gauge condition to ex-

hibit a quadratic or higher order zero. This implies that at this point in function

space, the gauge condition is satis¬ed by at least two gauge equivalent con¬g-

urations, i.e. vanishing of ∆f [A] implies the existence of zero modes associated

with M (135)

M χ0 = 0

54 F. Lenz

and therefore the gauge choice is ambiguous. The (connected) spaces of gauge

¬elds which make the gauge choice ambiguous

MH = A det M = 0

are called Gribov horizons [51]. Around Gribov horizons, pairs of in¬nitesimally

close gauge equivalent ¬elds exist which satisfy the gauge condition. If on the

other hand two gauge ¬elds satisfy the gauge condition and are separated by an

in¬nitesimal gauge transformation, these two ¬elds are separated by a Gribov

horizon. The region beyond the horizon thus contains gauge copies of ¬elds

inside the horizon. In general, one therefore needs additional conditions to select

exactly one representative of the gauge orbits. The structure of Gribov horizons

and of the space of ¬elds which contain no Gribov copies depends on the choice

of the gauge. Without specifying further the procedure, we associate an in¬nite

potential energy V[A] with every gauge copy of a con¬guration which already

has been taken into account, i.e. after gauge ¬xing, the action is supposed to

contain implicitly this potential energy

S[A] ’ S[A] ’ d4 x V[A]. (140)

With the above expression, and given a reasonable gauge choice, the generating

functional is written as an integral over gauge orbits and can serve as starting

point for further formal developments such as the canonical formalism [36] or

applications e.g. perturbation theory.

The occurrence of Gribov horizons points to a more general problem in the

gauge ¬xing procedure. Unlike in electrodynamics, global gauge conditions may

not exist in non-abelian gauge theories [52]. In other words, it may not be pos-

sible to formulate a condition which in the whole space of gauge ¬elds selects

exactly one representative. This di¬culty of imposing a global gauge condition

is similar to the problem of a global coordinate choice on e.g. S 2 . In this case,

one either has to resort to some patching procedure and use more than one set

of coordinates (like for the Wu“Yang treatment of the Dirac Monopole [53]) or

deal with singular ¬elds arising from these gauge ambiguities (Dirac Monopole).

Gauge singularities are analogous to the coordinate singularities on non-trivial

manifolds (azimuthal angle on north pole).

The appearance of Gribov-horizons poses severe technical problems in ana-

lytical studies of non-abelian gauge theories. Elimination of redundant variables

is necessary for proper de¬nition of the path-integral of in¬nitely many variables.

In the gauge ¬xing procedure it must be ascertained that every gauge orbit is

represented by exactly one ¬eld-con¬guration. Gribov horizons may make this

task impossible. On the other hand, one may regard the existence of global gauge

conditions in QED and its non-existence in QCD as an expression of a funda-

mental di¬erence in the structure of these two theories which ultimately could

be responsible for their vastly di¬erent physical properties.

Topological Concepts in Gauge Theories 55

7 Instantons

7.1 Vacuum Degeneracy

Instantons are solutions of the classical Yang“Mills ¬eld equations with dis-

tinguished topological properties [54]. Our discussion of instantons follows the

pattern of that of the Nielsen“Olesen vortex or the ™t Hooft“Polyakov monopole

and starts with a discussion of con¬gurations of vanishing energy (cf. [55,34,57]).

As follows from the Yang“Mills Hamiltonian (100) in the Weyl gauge (99), static

zero-energy solutions of the equations of motion (84) satisfy

E = 0, B = 0,

and therefore are pure gauges (93)

1

U (x)∇U † (x).

A= (141)

ig

In the Weyl gauge, pure gauges in electrodynamics are gradients of time-indepen-

dent scalar functions. In SU (2) Yang“Mills theory, the manifold of zero-energy

solutions is according to (141) given by the set of all U (x) ∈ SU (2). Since

topologically SU (2) ∼ S 3 (cf. (52)), each U (x) de¬nes a mapping from the base

space R3 to S 3 . We impose the requirement that at in¬nity, U (x) approaches a

unique value independent of the direction of x

U (x) ’ const. for |x| ’ ∞. (142)

Thereby, the con¬guration space becomes compact R3 ’ S 3 (cf. (47)) and pure

gauges de¬ne a map

U (x) : S 3 ’’ S 3 (143)

to which, according to (43), a winding number can be assigned. This winding

number counts how many times the 3-sphere of gauge transformations U (x) is

covered if x covers once the 3-sphere of the compacti¬ed con¬guration space.

Via the degree of the map (49) de¬ned by U (x), this winding number can be

calculated [42,56] and expressed in terms of the gauge ¬elds

g2 g

Aa ‚j Aa ’

d3 x abc

Aa A b A c .

nw = (144)

ijk i k i j k

16π 2 3

The expression on the right hand side yields an integer only if A is a pure gauge.

Examples of gauge transformations giving rise to non-trivial winding (hedgehog

solution for n = 1) are

x„

}

Un (x) = exp{iπn (145)

x2 + p2

with winding number nw = n (cf. (51) for verifying the asymptotic behav-

ior (142)). Gauge transformations which change the winding number nw are

56 F. Lenz

0 1 2 3 4

d3 xB[A]2 as a function of the

Fig. 9. Schematic plot of the potential energy V [A] =

winding number (144)

called large gauge transformations. Unlike small gauge transformations, they

cannot be deformed continuously to U = 1.

These topological considerations show that Yang“Mills theory considered as a

classical system possesses an in¬nity of di¬erent lowest energy (E = 0) solutions

which can be labeled by an integer n. They are connected to each other by

gauge ¬elds which cannot be pure gauges and which therefore produce a ¬nite

value of the magnetic ¬eld, i.e. of the potential energy. The schematic plot of the

potential energy in Fig. 9 shows that the ground state of QCD can be expected

to exhibit similar properties as that of a particle moving in a periodic potential.

In the quantum mechanical case too, an in¬nite degeneracy is present with the

winding number in gauge theories replaced by the integer characterizing the

equilibrium positions of the particle.

7.2 Tunneling

“Classical vacua” are states with values of the coordinate of a mechanical system

x = n given by the equilibrium positions. Correspondingly, in gauge theories the

classical vacua, the “n-vacua” are given by the pure gauges ((141) and (145)). To

proceed from here to a description of the quantum mechanical ground state, tun-

neling processes have to be included which, in such a semi-classical approxima-

tion, connect classical vacua with each other. Thereby the quantum mechanical

ground state becomes a linear superposition of classical vacua. Such tunneling

solutions are most easily obtained by changing to imaginary time with a con-

comitant change in the time component of the gauge potential

t ’ ’it , A0 ’ ’iA0 . (146)

The metric becomes Euclidean and there is no distinction between covariant and

contravariant indices. Tunneling solutions are solutions of the classical ¬eld equa-

tions derived from the Euclidean action SE , i.e. the Yang“Mills action (cf. (80))

modi¬ed by the substitution (146). We proceed in a by now familiar way and

Topological Concepts in Gauge Theories 57

derive the Bogomol™nyi bound for topological excitations in Yang“Mills theories.

To this end we rewrite the action (cf. (87))

1 1 1a

±Fµν Fµν + (Fµν “ Fµν )2

a ˜a ˜a

d4 x Fµν Fµν =

a a

d4 x

SE = (147)

4 4 2

1

≥± a ˜a

d4 x Fµν Fµν (148)

4

This bound for SE (Bogomol™nyi bound) is determined by the topological charge

ν , i.e. it can be rewritten as a surface term

g2 a ˜a

d4 x Fµν Fµν = d σµ K µ

ν= (149)

32π 2

of the topological current

g 2 µ±βγ g abc a b c

Aa ‚β Aa ’

Kµ = A± Aβ Aγ . (150)

± γ

16π 2 3

Furthermore, if we assume K to vanish at spatial in¬nity so that

+∞

d

K 0 d3 x = nw (t = ∞) ’ nw (t = ’∞) ,

ν= dt (151)

dt

’∞

the charge ν is seen to be quantized as a di¬erence of two winding numbers.

I ¬rst discuss the formal implications of this result. The topological charge

has been obtained as a di¬erence of winding numbers of pure (time-independent)

gauges (141) satisfying the condition (142). With the winding numbers, also ν

is a topological invariant. It characterizes the space-time dependent gauge ¬elds

Aµ (x). Another and more direct approach to the topological charge (149) is

provided by the study of cohomology groups. Cohomology groups characterize

connectedness properties of topological spaces by properties of di¬erential forms

and their integration via Stokes™ theorem (cf. Chap. 12 of [58] for an introduc-

tion).

Continuous deformations of gauge ¬elds cannot change the topological charge.

This implies that ν remains unchanged under continuous gauge transformations.

In particular, the ν = 0 equivalence class of gauge ¬elds containing Aµ = 0 as

an element cannot be connected to gauge ¬elds with non-vanishing topological

charge. Therefore, the gauge orbits can be labeled by ν. Field con¬gurations

with ν = 0 connect vacua (zero-energy ¬eld con¬gurations) with di¬erent wind-

ing number ((151) and (144)). Therefore, the solutions to the classical Euclidean

¬eld equations with non-vanishing topological charge are the tunneling solutions

needed for the construction of the semi-classical Yang“Mills ground state.

Like in the examples discussed in the previous sections, the ¬eld equations

simplify if the Bogomol™nyi bound is saturated. In the case of Yang“Mills the-

ory, the equations of motion can then be solved in closed form. Solutions with

topological charge ν = 1 (ν = ’1) are called instantons (anti-instantons). Their

action is given by

8π 2

SE = 2 .

g

58 F. Lenz

By construction, the action of any other ¬eld con¬guration with |ν| = 1 is larger.

Solutions with action SE = 8π 2 |ν|/g 2 for |ν| > 1 are called multi-instantons.

According to (147), instantons satisfy

Fµν = ±Fµν .

˜ (152)

The interchange Fµν ” Fµν corresponding in Minkowski space to the inter-

˜

change E ’ B, B ’ ’E is a duality transformation and ¬elds satisfying (152)

are said to be selfdual (+) or anti-selfdual (’) respectively. A spherical Ansatz

yields the solutions

192ρ4

2 ·aµν xν 1

Aa = ’ 2

Fµν = , (153)

µ

g x2 + ρ2 g 2 (x2 + ρ2 )4

with the ™t Hooft symbol [59]

±

µ, ν = 1, 2, 3

aµν

δaµ ν=0

·aµν =

’δaν µ=0 .

The size of the instanton ρ can be chosen freely. Asymptotically, gauge potential

and ¬eld strength behave as

1 1

A F .

’’ ’’

x4

|x|’∞ |x|’∞

x

The unexpectedly strong decrease in the ¬eld strength is the result of a partial

cancellation of abelian and non-abelian contributions to Fµν (78). For instantons,

the asymptotics of the gauge potential is actually gauge dependent. By a gauge

transformation, the asymptotics can be changed to x’3 . Thereby the gauge

¬elds develop a singularity at x = 0, i.e. in the center of the instanton. In this

“singular” gauge, the gauge potential is given by

2ρ2 ·aµν xν

¯

=’ 2 2 ·aµν = ·aµν (1 ’ 2δµ,0 )(1 ’ 2δν,0 ) .

Aa , ¯ (154)