where one integrates over paths satisfying the boundary condition x(β/2) =

x(’β/2) + π. A careful study of the trace operation in the case of periodic

potentials shows that x(’β/2) varies over only one period (see Appendix A).

Therefore, from (42), we derive the path integral representation

Z(θ, β) = [dx(t)] exp [’S(x) + i θ]

x(β/2)=x(’β/2)+ π

β/2

θ

[dx(t)] exp ’S(x) + i

= dt x(t) . (45)

™

π ’β/2

x(β/2)=x(’β/2) (mod π)

Note that is a topological number since two trajectories with di¬erent values

of cannot be related continuously. In the same way,

β/2

1

Q= dt x(t)

™

π ’β/2

is a topological charge; it depends on the trajectory only through the boundary

conditions.

For β large and g ’ 0, the path integral is dominated by the constant

solutions xc (t) = 0 mod π corresponding to the = 0 sector. A non-trivial

θ dependence can come only from instanton (non-constant ¬nite action saddle

points) contributions corresponding to quantum tunnelling. Note that, quite

generally,

2

dt x(t) ± sin x(t) ≥ 0 ’ S ≥ cos x(+∞) ’ cos x(’∞) /g.

™ (46)

Chiral Anomalies and Topology 201

The action (44) is ¬nite for β ’ ∞ only if x(±∞) = 0 mod π. The non-vanishing

value of the r.h.s. of (46) is 2/g. This minimum is reached for trajectories xc that

are solutions of

xc = ± sin xc ’ xc (t) = 2 arctan e±(t’t0 ) ,

™

and the corresponding classical action then is

S(xc ) = 2/g .

The instanton solutions belong to the = ±1 sector and connect two consec-

utive minima of the potential. They yield the leading contribution to barrier

penetration for g ’ 0. An explicit calculation yields

4

E0 (g) = Epert. (g) ’ √ e’2/g cos θ[1 + O(g)],

πg

where Epert. (g) is the sum of the perturbative expansion in powers of g.

5.2 Instantons and Anomaly: CP(N-1) Models

We now consider a set of two-dimensional ¬eld theories, the CP (N ’ 1) models,

where again instantons and topology play a role and the semi-classical vacuum

has a similar periodic structure. The new feature is the relation between the

topological charge and the two-dimensional chiral anomaly.

Here, we describe mainly the nature of the instanton solutions and refer the

reader to the literature for a more detailed analysis. Note that the explicit cal-

culation of instanton contributions in the small coupling limit in the CP (N ’ 1)

models, as well as in the non-abelian gauge theories discussed in Sect. 5.3, re-

mains to large extent an unsolved problem. Due to the scale invariance of the

classical theory, instantons depend on a scale (or size) parameter. Instanton con-

tributions then involve the running coupling constant at the instanton size. Both

families of theories are UV asymptotically free. Therefore, the running coupling

is small for small instantons and the semi-classical approximation is justi¬ed.

However, in the absence of any IR cut-o¬, the running coupling becomes large

for large instantons, and it is unclear whether a semi-classical approximation

remains valid.

The CP(N-1) Manifolds. We consider a N -component complex vector • of

unit length:

• · • = 1.

¯

This •-space is also isomorphic to the quotient space U (N )/U (N ’ 1). In addi-

tion, two vectors • and • are considered equivalent if

• ≡ • ” •± = eiΛ •± . (47)

This condition characterizes the symmetric space and complex Grassmannian

manifold U (N )/U (1)/U (N ’ 1). It is isomorphic to the manifold CP (N ’ 1)

202 J. Zinn-Justin

(for N ’ 1-dimensional Complex Projective), which is obtained from CN by the

equivalence relation

z± ≡ z± if z± = »z±

where » belongs to the Riemann sphere (compacti¬ed complex plane).

The CP(N-1) Models. A symmetric space admits a unique invariant metric

and this leads to a unique action with two derivatives, up to a multiplicative

factor. Here, one representation of the unique U (N ) symmetric classical action

is

1

S(•, Aµ ) = d2 x Dµ • · Dµ • ,

g

in which g is a coupling constant and Dµ the covariant derivative:

Dµ = ‚µ + iAµ .

The ¬eld Aµ is a gauge ¬eld for the U (1) transformations:

Aµ (x) = Aµ (x) ’ ‚µ Λ(x).

• (x) = eiΛ(x) •(x) , (48)

The action is obviously U (N ) symmetric and the gauge symmetry ensures the

equivalence (47).

Since the action contains no kinetic term for Aµ , the gauge ¬eld is not a

dynamical but only an auxiliary ¬eld that can be integrated out. The action is

quadratic in A and the gaussian integration results in replacing in the action Aµ

by the solution of the A-¬eld equation

Aµ = i• · ‚µ • ,

¯ (49)

where (5.2) has been used. After this substitution, the ¬eld • · ‚µ • acts as a

¯

composite gauge ¬eld.

For what follows, however, we ¬nd it more convenient to keep Aµ as an

independent ¬eld.

Instantons. To prove the existence of locally stable non-trivial minima of the

action, the following Bogomolnyi inequality can be used (note the analogy with

(46)):

2

d2 x |Dµ • “ i ≥ 0,

µν Dν •|

( µν being the antisymmetric tensor, 12 = 1). After expansion, the inequality

can be cast into the form

S(•) ≥ 2π|Q(•)|/g

with

i i

Q(•) = ’ d2 x Dµ • · Dν • = · Dν Dµ • .

d2 x µν •

¯ (50)

µν

2π 2π

Chiral Anomalies and Topology 203

Then,

= 1i = 1 Fµν ,

i µν Dν Dµ µν [Dν , Dµ ] (51)

2 2

where Fµν is the curvature:

Fµν = ‚µ Aν ’ ‚ν Aµ .

Therefore, using (5.2),

1

d2 x

Q(•) = µν Fµν . (52)

4π

The integrand is proportional to the two-dimensional abelian chiral anomaly

(33), and thus is a total divergence:

1

2 µν Fµν = ‚µ µν Aν .

Substituting this form into (52) and integrating over a large disc of radius R,

one obtains

1

Q(•) = lim dxµ Aµ (x). (53)

2π R’∞ |x|=R

Q(•) thus depends only on the behaviour of the classical solution for |x| large and

is a topological charge. Finiteness of the action demands that at large distances

Dµ • vanishes and, therefore,

Dµ • = 0 ’ [Dµ , Dν ]• = Fµν • = 0 .

Since • = 0, this equation implies that Fµν vanishes and, thus, that Aµ is a pure

gauge (and • a gauge transform of a constant vector):

1

Aµ = ‚µ Λ(x) ’ Q(•) = lim dxµ ‚µ Λ(x) . (54)

2π R’∞ |x|=R

The topological charge measures the variation of the angle Λ(x) on a large circle,

which is a multiple of 2π because • is regular. One is thus led to the consideration

of the homotopy classes of mappings from U (1), that is S1 to S1 , which are

characterized by an integer n, the winding number. This is equivalent to the

statement that the homotopy group π1 (S1 ) is isomorphic to the additive group

of integers Z.

Then,

Q(•) = n =’ S(•) ≥ 2π|n|/g .

The equality S(•) = 2π|n|/g corresponds to a local minimum and implies that

the classical solutions satisfy ¬rst order partial di¬erential (self-duality) equa-

tions:

Dµ • = ±i µν Dν • . (55)

For each sign, there is really only one equation, for instance µ = 1, ν = 2. It is

simple to verify that both equations imply the •-¬eld equations, and combined

204 J. Zinn-Justin

with the constraint (5.2), the A-¬eld equation (49). In complex coordinates z =

x1 + ix2 , z = x1 ’ ix2 , they can be written as

¯

‚z •± (z, z ) = ’iAz (z, z )•± (z, z ),

¯ ¯ ¯

‚z •± (z, z ) = ’iAz (z, z )•± (z, z ).

¯ ¯ ¯

¯ ¯

Exchanging the two equations just amounts to exchange • and •. Therefore, we

¯

solve only the second equation which yields

•± (z, z ) = κ(z, z )P± (z),

¯ ¯

where κ(z, z ) is a particular solution of

¯

‚z κ(z, z ) = ’iAz (z, z )κ(z, z ).

¯ ¯ ¯

¯ ¯

Vector solutions of (55) are proportional to holomorphic or anti-holomorphic

(depending on the sign) vectors (this re¬‚ects the conformal invariance of the

classical ¬eld theory). The function κ(z, z ), which gauge invariance allows to

¯

choose real (this corresponds to the ‚µ Aµ = 0 gauge), then is constrained by the

condition (5.2):

κ2 (z, z ) P · P = 1 .

¯

¯

The asymptotic conditions constrain the functions P± (z) to be polynomials.

Common roots to all P± would correspond to non-integrable singularities for •±

and, therefore, are excluded by the condition of ¬niteness of the action. Finally,

if the polynomials have maximal degree n, asymptotically

c±

P± (z) ∼ c± z n ’ •± ∼ √ (z/¯)n/2 .

z

c·c

¯

When the phase of z varies by 2π, the phase of •± varies by 2nπ, showing that

the corresponding winding number is n.

The Structure of the Semi-classical Vacuum. In contrast to our analysis of

periodic potentials in quantum mechanics, here we have discussed the existence

of instantons without reference to the structure of the classical vacuum. To

¬nd an interpretation of instantons in gauge theories, it is useful to express the

results in the temporal gauge A2 = 0. Then, the action is still invariant under

space-dependent gauge transformations. The minima of the classical • potential

correspond to ¬elds •(x1 ), where x1 is the space variable, gauge transforms of

a constant vector:

•(x1 ) = eiΛ(x1 ) v , v · v = 1 .

¯

Moreover, if the vacuum state is invariant under space re¬‚ection, •(+∞) =

•(’∞) and, thus,

Λ(+∞) ’ Λ(’∞) = 2νπ ν ∈ Z .

Again ν is a topological number that classi¬es degenerate classical minima, and

the semi-classical vacuum has a periodic structure. This analysis is consistent

Chiral Anomalies and Topology 205

with Gauss™s law, which implies only that states are invariant under in¬nitesimal

gauge transformations and, thus, under gauge transformations of the class ν = 0

that are continuously connected to the identity.

We now consider a large rectangle with extension R in the space direction

and T in the euclidean time direction and by a smooth gauge transformation

continue the instanton solution to the temporal gauge. Then, the variation of

the pure gauge comes entirely from the sides at ¬xed time. For R ’ ∞, one

¬nds