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 .

4π

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