The two contributions are now separately convergent. When m ’ 0, the m2

factor dominates the logarithmic IR divergence and the contribution vanishes.

In the second term, in the limit M ’ ∞, one obtains

d2 q

1 e

∼ ’4eM µν kν =’

2

Cµ (k)|m’0 ,M ’∞ µν kν ,

(2π)2 (q 2 + M 2 )2 π

in agreement with (33).

4.4 Non-Abelian Vector Gauge Fields and Abelian Axial Current

We still consider an abelian axial current but now in the framework of a non-

abelian gauge theory. The fermion ¬elds transform non-trivially under a gauge

group G and Aµ is the corresponding gauge ¬eld. The action is

¯ ¯

S(ψ, ψ; A) = ’ d4 x ψ(x)Dψ(x)

with the convention (9) and

D = ‚ +A. (34)

196 J. Zinn-Justin

In a gauge transformation represented by a unitary matrix g(x), the gauge ¬eld

Aµ and the Dirac operator become

Aµ (x) ’ g(x)‚µ g’1 (x) + g(x)Aµ (x)g’1 (x) ’ D ’ g’1 (x)Dg(x) . (35)

The axial current

¯

5

Jµ (x) = iψ(x)γ5 γµ ψ(x)

is still gauge invariant. Therefore, no new calculation is needed; the result is com-

pletely determined by dimensional analysis, gauge invariance, and the preceding

abelian calculation that yields the term of order A2 :

i

‚» J» (x) = ’

5

tr Fµν Fρσ , (36)

µνρσ

16π 2

in which Fµν now is the corresponding curvature (10). Again this expression

must be a total derivative. Indeed, one veri¬es that

2

tr Fµν Fρσ = 4 µνρσ ‚µ tr(Aν ‚ρ Aσ + Aν Aρ Aσ ). (37)

µνρσ

3

4.5 Anomaly and Eigenvalues of the Dirac Operator

We assume that the spectrum of D, the Dirac operator in a non-abelian gauge

¬eld (34), is discrete (putting temporarily the fermions in a box if necessary)

and call dn and •n (x) the corresponding eigenvalues and eigenvectors:

D•n = dn •n .

For a unitary or orthogonal group, the massless Dirac operator is anti-hermitian;

therefore, the eigenvalues are imaginary and the eigenvectors orthogonal. In ad-

dition, we choose them with unit norm.

The eigenvalues are gauge invariant because, in a gauge transformation char-

acterized by a unitary matrix g(x), the Dirac operator transforms like in (35),

and thus simply

•n (x) ’ g(x)•n (x).

The anticommutation Dγ5 + γ5D = 0 implies

Dγ5 •n = ’dn γ5 •n .

Therefore, either dn is di¬erent from zero and γ5 •n is an eigenvector of D with

eigenvalue ’dn , or dn vanishes. The eigenspace corresponding to the eigenvalue

0 then is invariant under γ5 , which can be diagonalized: the eigenvectors of D

can be chosen eigenvectors of de¬nite chirality, that is eigenvectors of γ5 with

eigenvalue ±1:

D•n = 0 , γ5 •n = ±•n .

We call n+ and n’ the dimensions of the eigenspace of positive and negative

chirality, respectively.

Chiral Anomalies and Topology 197

We now consider the determinant of the operator D + m regularized by mode

truncation (mode regularization):

detN (D + m) = (dn + m),

n¤N

keeping the N lowest eigenvalues of D (in modulus), with N ’ n+ ’ n’ even, in

such a way that the corresponding subspace remains γ5 invariant.

The regularization is gauge invariant because the eigenvalues of D are gauge

invariant.

Note that in the truncated space

tr γ5 = n+ ’ n’ . (38)

The trace of γ5 equals n+ ’ n’ , the index of the Dirac operator D. A non-

vanishing index thus endangers axial current conservation.

In a chiral transformation (20) with constant θ, the determinant of (D + m)

becomes

detN (D + m) ’ detN eiθγ5 (D + m)eiθγ5 .

We now consider the various eigenspaces.

If dn = 0, the matrix γ5 is represented by the Pauli matrix σ1 in the sum of

eigenspaces corresponding to the two eigenvalues ±dn and D + m by dn σ3 + m.

The determinant in the subspace then is

det eiθσ1 (dn σ3 + m)eiθσ1 = det e2iθσ1 det(dn σ3 + m) = m2 ’ d2 ,

n

because σ1 is traceless.

In the eigenspace of dimension n+ of vanishing eigenvalues dn with eigenvec-

tors with positive chirality, γ5 is diagonal with eigenvalue 1 and, thus,

mn+ ’ mn+ e2iθn+ .

Similarly, in the eigenspace of chirality ’1 and dimension n’ ,

mn’ ’ mn’ e’2iθn’ .

We conclude

detN eiθγ5 (D + m)eiθγ5 = e2iθ(n+ ’n’ ) detN (D + m),

The ratio of the two determinants is independent of N . Taking the limit N ’ ∞,

one ¬nds

’1

= e2iθ(n+ ’n’ ) .

det eiγ5 θ (D + m)eiγ5 θ (D + m) (39)

Note that the l.h.s. of (39) is obviously 1 when θ = nπ, which implies that the

coe¬cient of 2θ in the r.h.s. must indeed be an integer.

The variation of ln det(D + m):

’1

= 2iθ (n+ ’ n’ ) ,

eiγ5 θ (D + m)eiγ5 θ (D + m)

ln det

198 J. Zinn-Justin

at ¬rst order in θ, is related to the variation of the action (18) (see (29)) and, thus,

to the expectation value of the integral of the divergence of the axial current,

d4 x ‚µ Jµ (x) in four dimensions. In the limit m = 0, it is thus related to the

5

space integral of the chiral anomaly (36).

We have thus found a local expression giving the index of the Dirac operator:

1

’ d4 x tr Fµν Fρσ = n+ ’ n’ . (40)

µνρσ

32π 2

Concerning this result several comments can be made:

(i) At ¬rst order in θ, in the absence of regularization, we have calculated

(ln det = tr ln)

ln det 1 + iθ γ5 + (D + m)γ5 (D + m)’1 ∼ 2iθ tr γ5 ,

where the cyclic property of the trace has been used. Since the trace of the matrix

γ5 in the full space vanishes, one could expect, naively, a vanishing result. But

trace here means trace in matrix space and in coordinate space and γ5 really

stands for γ5 δ(x ’ y). The mode regularization gives a well-de¬ned ¬nite result

for the ill-de¬ned product 0 — δ d (0).

(ii) The property that the integral (40) is quantized shows that the form of the

anomaly is related to topological properties of the gauge ¬eld since the integral

does not change when the gauge ¬eld is deformed continuously. The integral of

the anomaly over the whole space, thus, depends only on the behaviour at large

distances of the curvature tensor Fµν and the anomaly must be a total derivative

as (37) con¬rms.

(iii) One might be surprised that det D is not invariant under global chiral

transformations. However, we have just established that when the integral of the

anomaly does not vanish, det D vanishes. This explains that, to give a meaning

to the r.h.s. of (39), we have been forced to introduce a mass to ¬nd a non-trivial

result. The determinant of D in the subspace orthogonal to eigenvectors with

vanishing eigenvalue, even in presence of a mass, is chiral invariant by parity

doubling. But for n+ = n’ , this is not the case for the determinant in the

eigenspace of eigenvalue zero because the trace of γ5 does not vanish in this

eigenspace (38). In the limit m ’ 0, the complete determinant vanishes but not

the ratio of determinants for di¬erent values of θ because the powers of m cancel.

(iv) The discussion of the index of the Dirac operator is valid in any even

dimension. Therefore, the topological character and the quantization of the space

integral of the anomaly are general.

Instantons, Anomalies, and θ-Vacua

5

We now discuss the role of instantons in several examples where the classical

potential has a periodic structure with an in¬nite set of degenerate minima. We

exhibit their topological character, and in the presence of gauge ¬elds relate

Chiral Anomalies and Topology 199

them to anomalies and the index of the Dirac operator. Instantons imply that

the eigenstates of the hamiltonian depend on an angle θ. In the quantum ¬eld

theory the notion of θ-vacuum emerges.

5.1 The Periodic Cosine Potential

As a ¬rst example of the role of instantons when topology is involved, we consider

a simple hamiltonian with a periodic potential

g 1

2

H = ’ (d /dx ) + sin2 x . (41)

2 2g

The potential has an in¬nite number of degenerate minima for x = nπ, n ∈ Z.

Each minimum is an equivalent starting point for a perturbative calculation of

the eigenvalues of H. Periodicity implies that the perturbative expansions are

identical to all orders in g, a property that seems to imply that the quantum

hamiltonian has an in¬nite number of degenerate eigenstates. In reality, we know

that the exact spectrum of the hamiltonian H is not degenerate, due to barrier

penetration. Instead, it is continuous and has, at least for g small enough, a band

structure.

The Structure of the Ground State. To characterize more precisely the

structure of the spectrum of the hamiltonian (41), we introduce the operator T

that generates an elementary translation of one period π:

T ψ(x) = ψ(x + π).

Since T commutes with the hamiltonian,

[T, H] = 0 ,

both operators can be diagonalized simultaneously. Because the eigenfunctions

of H must be bounded at in¬nity, the eigenvalues of T are pure phases. Each

eigenfunction of H thus is characterized by an angle θ (pseudo-momentum)

associated with an eigenvalue of T :

T |θ = eiθ |θ .

The corresponding eigenvalues En (θ) are periodic functions of θ and, for g ’ 0,

are close to the eigenvalues of the harmonic oscillator:

En (θ) = n + 1/2 + O(g).

To all orders in powers of g, En (θ) is independent of θ and the spectrum of

H is in¬nitely degenerate. Additional exponentially small contributions due to

barrier penetration lift the degeneracy and introduce a θ dependence. To each

value of n then corresponds a band when θ varies in [0, 2π].

200 J. Zinn-Justin

Path Integral Representation. The spectrum of H can be extracted from

the calculation of the quantity

∞

1

’βH

dθ ei θ e’βEn (θ) .

Z (β) = tr T e =

2π n=0

Indeed,

e’βEn (θ) ,

Z(θ, β) ≡ ei θ Z (β) = (42)

n

where Z(θ, β) is the partition function restricted to states with a ¬xed θ angle.

The path integral representation of Z (β) di¬ers from the representation of

the partition function Z0 (β) only by the boundary conditions. The operator T

has the e¬ect of translating the argument x in the matrix element x | tr e’βH |x

before taking the trace. It follows that

Z (β) = [dx(t)] exp [’S(x)] , (43)

β/2

1

S(x) = x2 (t) + sin2 x(t)

™ dt , (44)

2g ’β/2