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Λ(+∞, 0) ’ Λ(’∞, 0) ’ [Λ(+∞, T ) ’ Λ(’∞, T )] = 2nπ .
Therefore, instantons interpolate between di¬erent classical minima. Like in the
case of the cosine potential, to project onto a proper quantum eigenstate, the “θ-
vacuum” corresponding to an angle θ, one adds, in analogy with the expression
(45), a topological term to the classical action. Here,
θ
S(•) ’ S(•) + i d2 x µν Fµν .


Remark. Replacing in the topological charge Q the gauge ¬eld by the explicit
expression (49), one ¬nds
i i
· ‚ν • = d•± § d•± ,
d2 x
Q(•) = µν ‚µ •
¯ ¯
2π 2π
where the notation of exterior di¬erential calculus has been used. We recognize
the integral of a two-form, a symplectic form, and 4πQ is the area of a 2-surface
embedded in CP (N ’ 1). A symplectic form is always closed. Here it is also
exact, so that Q is the integral of a one-form (cf. (53)):
i i
(•± d•± ’ •± d•± ) .
Q(•) = •± d•± =
¯ ¯ ¯
2π 4π

The O(3) Non-Linear σ-Model. The CP (1) model is locally isomorphic to
the O(3) non-linear σ-model, with the identi¬cation

φi (x) = •± (x)σ±β •β (x) ,
i
¯

where σ i are the three Pauli matrices.
Using, for example, an explicit representation of Pauli matrices, one indeed
veri¬es
φi (x)φi (x) = 1 , ‚µ φi (x)‚µ φi (x) = 4Dµ • · Dµ • .
Therefore, the ¬eld theory can be expressed in terms of the ¬eld φi and takes the
form of the non-linear σ-model. The ¬elds φ are gauge invariant and the whole
physical picture is a picture of con¬nement of the charged scalar “quarks” •± (x)
and the propagation of neutral bound states corresponding to the ¬elds φi .
Instantons in the φ description take the form of φ con¬gurations with uniform
limit for |x| ’ ∞. Thus, they de¬ne a mapping from the compacti¬ed plane
206 J. Zinn-Justin

topologically equivalent to S2 to the sphere S2 (the φi con¬gurations). Since
π2 (S2 ) = Z, the • and φ pictures are consistent.
In the example of CP (1), a solution of winding number 1 is
1 z
•1 = √ •2 = √
, .
1 + zz
¯ 1 + zz
¯

Translating the CP (1) minimal solution into the O(3) σ-model language, one
¬nds
1 z’z 1 ’ zz
z+z ¯ ¯ ¯
φ1 = , φ2 = , φ3 = .
1 + zz
¯ i 1 + zz
¯ 1 + zz
¯
This de¬nes a stereographic mapping of the plane onto the sphere S2 , as one
veri¬es by setting z = tan(·/2)eiθ , · ∈ [0, π].
In the O(3) representation

i 1 1
d•± § d•± = φi dφj § φk ≡ d2 x φi ‚µ φj ‚ν φk .
Q= ¯ ijk µν ijk
2π 8π 8π
The topological charge 4πQ has the interpretation of the area of the sphere S2 ,
multiply covered, and embedded in R3 . Its value is a multiple of the area of S2 ,
which in this interpretation explains the quantization.


5.3 Instantons and Anomaly: Non-Abelian Gauge Theories

We now consider non-abelian gauge theories in four dimensions. Again, gauge
¬eld con¬gurations can be found that contribute to the chiral anomaly and
for which, therefore, the r.h.s. of (40) does not vanish. A specially interesting
example is provided by instantons, that is ¬nite action solutions of euclidean
¬eld equations.
To discuss this problem it is su¬cient to consider pure gauge theories and the
gauge group SU (2), since a general theorem states that for a Lie group containing
SU (2) as a subgroup the instantons are those of the SU (2) subgroup.
In the absence of matter ¬elds it is convenient to use a SO(3) notation. The
gauge ¬eld Aµ is a SO(3) vector that is related to the element Aµ of the Lie
algebra used previously as gauge ¬eld by

Aµ = ’ 1 iAµ · σ ,
2

where σi are the three Pauli matrices. The gauge action then reads
1 2
S(Aµ ) = [Fµν (x)] d4 x ,
4g 2

(g is the gauge coupling constant) where the curvature

Fµν = ‚µ Aν ’ ‚ν Aµ + Aµ — Aν ,

is also a SO(3) vector.
Chiral Anomalies and Topology 207

The corresponding classical ¬eld equations are
Dν Fνµ = ‚ν Fνµ + Aν — Fνµ = 0 . (56)
The existence and some properties of instantons in this theory follow from con-
siderations analogous to those presented for the CP (N ’ 1) model.
We de¬ne the dual of the tensor Fµν by
˜ 1
Fµν = 2 µνρσ Fρσ .
Then, the Bogomolnyi inequality
2
d x Fµν (x) ± Fµν (x) ≥0
˜
4



implies
S(Aµ ) ≥ 8π 2 |Q(Aµ )|/g 2
with
1
d4 x Fµν · Fµν .
˜
Q(Aµ ) = (57)
32π 2
The expression Q(Aµ ) is proportional to the integral of the chiral anomaly (36),
here written in SO(3) notation.
We have already pointed out that the quantity Fµν · Fµν is a pure divergence
˜
(37):
Fµν · Fµν = ‚µ Vµ
˜
with
2
Vµ = ’4 tr Aν ‚ρ Aσ + Aν Aρ Aσ
µνρσ
3
Aν · ‚ρ Aσ + 1 Aν · (Aρ — Aσ ) .
=2 (58)
µνρσ 3

The integral thus depends only on the behaviour of the gauge ¬eld at large
distances and its values are quantized (40). Here again, as in the CP (N ’ 1)
model, the bound involves a topological charge: Q(Aµ ).
Stokes theorem implies

d4 x ‚µ Vµ = d„¦ nµ Vµ ,
ˆ
D ‚D

where d„¦ is the measure on the boundary ‚D of the four-volume D and nµ the
ˆ
unit vector normal to ‚D. We take for D a sphere of large radius R and ¬nd for
the topological charge
1 1
d4 x tr Fµν · Fµν =
˜ R3
Q(Aµ ) = d„¦ nµ Vµ ,
ˆ (59)
32π 2 32π 2 r=R

The ¬niteness of the action implies that the classical solution must asymptoti-
cally become a pure gauge, that is, with our conventions,
Aµ = ’ 1 iAµ · σ = g(x)‚µ g’1 (x) + O |x|’2 |x| ’ ∞ . (60)
2
208 J. Zinn-Justin

The element g of the SU (2) group can be parametrized in terms of Pauli matri-
ces:
g = u4 1 + iu · σ , (61)
where the four-component real vector (u4 , u) satis¬es

u2 + u2 = 1 ,
4

and thus belongs to the unit sphere S3 . Since SU (2) is topologically equivalent to
the sphere S3 , the pure gauge con¬gurations on a sphere of large radius |x| = R
de¬ne a mapping from S3 to S3 . Such mappings belong to di¬erent homotopy
classes that are characterized by an integer called the winding number. Here, we
identify the homotopy group π3 (S3 ), which again is isomorphic to the additive
group of integers Z.
The simplest one to one mapping corresponds to an element of the form
x4 1 + ix · σ
r = (x2 + x2 )1/2
g(x) = , 4
r
and thus
’2
Ai = ’2xi r’2 .
Ai ∼ 2 (x4 δim + imk xk ) r ,
m 4
r’∞

Note that the transformation

g(x) ’ U1 g(x)U† = g(Rx),
2

where U1 and U2 are two constant SU (2) matrices, induces a SO(4) rotation
of matrix R of the vector xµ . Then,

U2 ‚µ g† (x)U† = Rµν ‚ν g† (Rx), U1 g(x)‚µ g† (x)U† = g(Rx)Rµν ‚ν g† (Rx)
1 1

and, therefore,
U1 Aµ (x)U† = Rµν Aν (Rx).
1

Introducing this relation into the de¬nition (58) of Vµ , one veri¬es that the
dependence on the matrix U1 cancels in the trace and, thus, Vµ transforms like
a 4-vector. Since only one vector is available, and taking into account dimensional
analysis, one concludes that
Vµ ∝ xµ /r4 .
For r ’ ∞, Aµ approaches a pure gauge (60) and, therefore, Vµ can be
transformed into
1
Vµ ∼ ’ µνρσ Aν · (Aρ — Aσ ).
3
r’∞

It is su¬cient to calculate V1 . We choose ρ = 3, σ = 4 and multiply by a factor
six to take into account all other choices. Then,

V1 ∼ 16 6
= 16x1 /r4
ijk (x4 δ2i + i2l xl )(x4 δ3j + j3m xm )xk /r
r’∞

and, thus,
Vµ ∼ 16xµ /r4 = 16ˆ µ /R3 .
n
Chiral Anomalies and Topology 209

The powers of R in (59) cancel and since d„¦ = 2π 2 , the value of the topological
charge is simply
Q(Aµ ) = 1 .
Comparing this result with (40), we see that we have indeed found the minimal
action solution.
Without explicit calculation we know already, from the analysis of the index
of the Dirac operator, that the topological charge is an integer:
1
d4 x Fµν · Fµν = n ∈ Z .
˜
Q(Aµ ) =
32π 2
As in the case of the CP (N ’1) model, this result has a geometric interpretation.
In general, in the parametrization (61),
8
Vµ ∼ µνρσ ±βγδ u± ‚ν uβ ‚ρ uγ ‚σ uδ .
r’∞ 3

A few algebraic manipulations starting from
1
Vµ duν § duρ § duσ ,
R3 d„¦ nµ Vµ =
ˆ µνρσ
6
S3

then yield
1
uµ duν § duρ § duσ ,
Q= (62)
µνρσ
12π 2

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. 45
( 78 .)



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