gx x + ρ2

7.3 Fermions in Topologically Non-trivial Gauge Fields

Fermions are severely a¬ected by the presence of gauge ¬elds with non-trivial

topological properties. A dynamically very important phenomenon is the appear-

ance of fermionic zero modes in certain gauge ¬eld con¬gurations. For a variety

of low energy hadronic properties, the existence of such zero modes appears to be

fundamental. Here I will not enter a detailed discussion of non-trivial fermionic

properties induced by topologically non-trivial gauge ¬elds. Rather I will try to

indicate the origin of the induced topological fermionic properties in the context

of a simple system. I will consider massless fermions in 1+1 dimensions moving

in an external (abelian) gauge ¬eld. The Lagrangian of this system is (cf. (81))

1 ¯

LY M = ’ F µν Fµν + ψiγ µ Dµ ψ, (155)

4

Topological Concepts in Gauge Theories 59

with the covariant derivative Dµ given in (5) and ψ denoting a 2-component

spinor. The Dirac algebra of the γ matrices

{γ µ , γ ν } = g µν

can be satis¬ed by the following choice in terms of Pauli-matrices (cf. (50))

γ 0 = „ 1 , γ 1 = i„ 2 , γ 5 = ’γ 0 γ 1 = „ 3 .

In Weyl gauge, A0 = 0, the Hamiltonian density (cf. (101)) is given by

12

E + ψ † Hf ψ ,

H= (156)

2

with

Hf = (i‚1 ’ eA1 ) γ 5 . (157)

The application of topological arguments is greatly simpli¬ed if the spectrum of

the fermionic states is discrete. We assume the ¬elds to be de¬ned on a circle

and impose antiperiodic boundary conditions for the fermions

ψ(x + L) = ’ψ(x) .

The (residual) time-independent gauge transformations are given by (6) and (7)

with the Higgs ¬eld φ replaced by the fermion ¬eld ψ. On a circle, the gauge

functions ±(x) have to satisfy (cf. (6))

2nπ

±(x + L) = ±(x) + .

e

The winding number nw of the mapping

U : S1 ’ S1

partitions gauge transformations into equivalence classes with representatives

given by the gauge functions

2πn

±n (x) = dn x, dn = . (158)

eL

Large gauge transformations de¬ne pure gauges

1

‚1 U † (x) ,

A1 = U (x) (159)

ie

which inherit the winding number (cf. (144)). For 1+1 dimensional electrody-

namics the winding number of a pure gauge is given by

L

e

nw = ’ dxA1 (x) . (160)

2π 0

60 F. Lenz

As is easily veri¬ed, eigenfunctions and eigenvalues of Hf are given by

2π 1

x

ψn (x) = e’ie En (a) = ± (n + ’ a) ,

A1 dx’iEn (a)x

u± , (161)

0

L 2

with the positive and negative chirality eigenspinors u± of „ 3 and the zero mode

of the gauge ¬eld

L

e

a= dxA1 (x) .

2π 0

We now consider a change of the external gauge ¬eld A1 (x) from A1 (x) = 0 to

a pure gauge of winding number nw . The change is supposed to be adiabatic,

such that the fermions can adjust at each instance to the changed value of the

external ¬eld. In the course of this change, a changes continuously from 0 to nw .

Note that adiabatic changes of A1 generate ¬nite ¬eld strengths and therefore

do not correspond to gauge transformations. As a consequence we have

En (nw ) = En’nw (0). (162)

As expected, no net change of the spectrum results from this adiabatic changes

between two gauge equivalent ¬elds A1 . However, in the course of these changes

the labeling of the eigenstates has changed. nw negative eigenenergies of a certain

chirality have become positive and nw positive eigenenergies of the opposite

chirality have become negative. This is called the spectral ¬‚ow associated with

this family of Dirac operators. The spectral ¬‚ow is determined by the winding

number of pure gauges and therefore a topological invariant. The presence of

pure gauges with non-trivial winding number implies the occurrence of zero

modes in the process of adiabatically changing the gauge ¬eld. In mathematics,

the existence of zero modes of Dirac operators has become an important tool in

topological investigations of manifolds ([60]). In physics, the spectral ¬‚ow of the

Dirac operator and the appearance of zero modes induced by topologically non-

trivial gauge ¬elds is at the origin of important phenomena like the formation

of condensates or the existence of chiral anomalies.

7.4 Instanton Gas

In the semi-classical approximation, as sketched above, the non-perturbative

QCD ground state is assumed to be given by topologically distinguished pure

gauges and the instantons connecting the di¬erent classical vacuum con¬gura-

tions. In the instanton model for the description of low-energy strong interaction

physics, one replaces the QCD partition function (134), i.e. the weighted sum

over all gauge ¬elds by a sum over (singular gauge) instanton ¬elds (154)

N

U (i) Aµ (i) U + (i) ,

Aµ = (163)

i=1

with

xν ’ zν (i)

ρ2

Aµ (i) = ’¯aµν „a .

·

g[x ’ z(i)]2 [x ’ z(i)]2 + ρ2

Topological Concepts in Gauge Theories 61

The gauge ¬eld is composed of N instantons with their centers located at the

positions z(i) and color orientations speci¬ed by the SU (2) matrices U (i). The

instanton ensemble for calculation of n’point functions is obtained by summing

over these positions and color orientations

N

d4 x J·A

[dU (i)dz(i)] e’SE [A]+i

Z[J] = .

i=1

Starting point of hadronic phenomenology in terms of instantons are the fermi-

onic zero modes induced by the non-trivial topology of instantons. The zero

modes are concentrated around each individual instanton and can be constructed

in closed form

Dψ0 = 0,

/

ρ γx 1 + γ5

√

ψ0 = •0 ,

3

2

π x2 (x2 + ρ2 ) 2

where •0 is an appropriately chosen constant spinor. In the instanton model, the

functional integration over quarks is truncated as well and replaced by a sum

over the zero modes in a given con¬guration of non-overlapping instantons. A

successful description of low-energy hadronic properties has been achieved [61]

although a dilute gas of instantons does not con¬ne quarks and gluons. It appears

that the low energy-spectrum of QCD is dominated by the chiral properties of

QCD which in turn seem to be properly accounted for by the instanton induced

fermionic zero modes. The failure of the instanton model in generating con¬ne-

ment will be analyzed later and related to a de¬cit of the model in properly

accounting for the ˜center symmetry™ in the con¬ning phase.

To describe con¬nement, merons have been proposed [62] as the relevant

¬eld con¬gurations. Merons are singular solutions of the classical equations of

motion [63]. They are literally half-instantons, i.e. up to a factor of 1/2 the meron

gauge ¬elds are identical to the instanton ¬elds in the “regular gauge” (153)

1 aI 1 ·aµν xν

Aµ (x) = ’

Aa M (x) = ,

µ

x2

2 g

and carry half a unit of topological charge. By this change of normalization, the

cancellation between abelian and non-abelian contributions to the ¬eld strength

is upset and therefore, asymptotically

1 1

A∼ F∼

, .

x2

x

The action

1

S∼ d4 x ,

x4

exhibits a logarithmic infrared divergence in addition to the ultraviolet diver-

gence. Unlike instantons in singular gauge (A ∼ x’3 ), merons always overlap. A

dilute gas limit of an ensemble of merons does not exist, i.e. merons are strongly

62 F. Lenz

interacting. The absence of a dilute gas limit has prevented development of

a quantitative meron model of QCD. Recent investigations [64] in which this

strongly interacting system of merons is treated numerically indeed suggest that

merons are appropriate e¬ective degrees of freedom for describing the con¬ning

phase.

7.5 Topological Charge and Link Invariants

Because of its wide use in the topological analysis of physical systems, I will

discuss the topological charge and related topological invariants in the concluding

paragraph on instantons.

The quantization of the topological charge ν is a characteristic property of

the Yang“Mills theory in 4 dimensions and has its origin in the non-triviality of

the mapping (143). Quantities closely related to ν are of topological relevance in

other ¬elds of physics. In electrodynamics topologically non-trivial gauge trans-

formations in 3 space dimensions do not exist π3 (S 1 ) = 0 and therefore the

topological charge is not quantized. Nevertheless, with

˜

K0 = 0ijk

Ai ‚j Ak ,

the charge

d3 x A · B

˜

d3 xK 0 =

hB = (164)

describes topological properties of ¬elds. For illustration we consider two linked

magnetic ¬‚ux tubes (Fig. 10) with the axes of the ¬‚ux tubes forming closed

curves C1,2 . Since hB is gauge invariant (the integrand is not, but the integral

over the scalar product of the transverse magnetic ¬eld and the (longitudinal)

change in the gauge ¬eld vanishes), we may assume the vector potential to satisfy

the Coulomb gauge condition

divA = 0 ,

which allows us to invert the curl operator

1

(∇— )’1 = ’∇ — (165)

∆

C2

C1

Fig. 10. Linked magnetic ¬‚ux tubes

Topological Concepts in Gauge Theories 63

˜

and to express K 0 uniquely in terms of the magnetic ¬eld

x ’x

1 1

K0 = ’ ∇ — B · B = d3 x B(x) — B(x ) ·

˜ d3 x .

|x ’ x|3

∆ 4π

For single ¬eld lines,

ds1 ds2

δ(x ’ s1 (t)) + b2 δ(x ’ s2 (t))

B(x) = b1

dt dt

the above integral is given by the linking number of the curves C1,2 (cf. (1)).

Integrating ¬nally over the ¬eld lines, the result becomes proportional to the

magnetic ¬‚uxes φ1,2

hB = 2 φ1 φ2 lk{C1 , C2 } . (166)

This result indicates that the charge hB , the “magnetic helicity”, is a topolog-

ical invariant. For an arbitrary magnetic ¬eld, the helicity hB can be interpreted

as an average linking number of the magnetic ¬eld lines [22]. The helicity hω of

vector ¬elds has actually been introduced in hydrodynamics [5] with the vector

potential replaced by the velocity ¬eld u of a ¬‚uid and the magnetic ¬eld by

the vorticity ω = ∇ — u. The helicity measures the alignment of velocity and

vorticity. The prototype of a “helical” ¬‚ow [65] is

1