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µ
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

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