gauge ¬eld. The action takes the form

¯ ¯

S(ψ, ψ; A) = ’ d4 x ψ(x) 1 (1 + γ5 ) D ψ(x)

2

(the discussion with 1 (1 ’ γ5 ) is similar).

2

The gauge theory is consistent only if the partition function

¯ ¯

Z(Aµ ) = dψdψ exp ’S(ψ, ψ; A)

is gauge invariant.

We introduce the generators t± of the gauge group in the fermion represen-

tation and de¬ne the corresponding current by

¯

Jµ (x) = ψ 1 (1 + γ5 ) γµ t± ψ .

±

2

Again, the invariance of Z(Aµ ) under an in¬nitesimal gauge transformation

implies for the current Jµ = Jµ t± the covariant conservation equation

±

Dµ Jµ = 0

with

Dµ = ‚µ + [Aµ , •].

The calculation of the quadratic contribution to the anomaly is simple: the ¬rst

regularization adopted for the calculation in Sect. 4.2 is also suited to the present

situation since the current-gauge ¬eld three-point function is symmetric in the

external arguments. The group structure is re¬‚ected by a simple geometric factor.

The global factor can be taken from the abelian calculation. It di¬ers from result

(26) by a factor 1/2 that comes from the projector 1 (1 + γ5 ). The general form

2

of the term of degree 3 in the gauge ¬eld can also easily be found while the

calculation of the global factor is somewhat tedious. We show in Sect. 6.3 that

it can be obtained from consistency conditions. The complete expression then

reads

1

±

(Dµ Jµ (x)) = ’ tr t± Aν ‚ρ Aσ + 1 Aν Aρ Aσ

‚µ . (65)

µνρσ 2

24π 2

If the projector 1 (1 + γ5 ) is replaced by 1 (1 ’ γ5 ), the sign of the anomaly

2 2

changes.

Unless the anomaly vanishes identically, there is an obstruction to the con-

struction of the gauge theory. The ¬rst term is proportional to

tr t± tβ tγ + tγ tβ

1

d±βγ = .

2

The second term involves the product of four generators, but taking into account

the antisymmetry of the tensor, one product of two consecutive can be replaced

by a commutator. Therefore, the term is also proportional to d±βγ .

Chiral Anomalies and Topology 215

For a unitary representation the generators t± are, with our conventions,

antihermitian. Therefore, the coe¬cients d±βγ are purely imaginary:

†

d— = = ’d±βγ .

tr t± tβ tγ + tγ tβ

1

±βγ 2

These coe¬cients vanish for all representations that are real: the t± antisymmet-

ric, or pseudo-real, that is t± = ’S Tt± S ’1 . It follows that the only non-abelian

groups that can lead to anomalies in four dimensions are SU (N ) for N ≥ 3,

SO(6), and E6 .

6.3 Wess“Zumino Consistency Conditions

In Sect. 6.2, we have calculated the part of the anomaly that is quadratic in the

gauge ¬eld and asserted that the remaining non-quadratic contributions could

be obtained from geometric arguments. The anomaly is the variation of a func-

tional under an in¬nitesimal gauge transformation. This implies compatibility

conditions, which here are constraints on the general form of the anomaly, the

Wess“Zumino consistency conditions. One convenient method to derive these

constraints is based on BRS transformations: one expresses that BRS transfor-

mations are nilpotent.

In a BRS transformation, the variation of the gauge ¬eld Aµ takes the form

δBRS Aµ (x) = Dµ C(x)¯ ,

µ (66)

where C is a fermion spinless “ghost” ¬eld and µ an anticommuting constant.

¯

The corresponding variation of ln Z(Aµ ) is

δBRS ln Z(Aµ ) = ’ d4 x Jµ (x) Dµ C(x)¯ .

µ (67)

The anomaly equation has the general form

Dµ Jµ (x) = A (Aµ ; x) .

In terms of A, the equation (67), after an integration by parts, can be rewritten

as

δBRS ln Z(Aµ ) = d4 x A (Aµ ; x) C(x)¯ .µ

Since the r.h.s. is a BRS variation, it satis¬es a non-trivial constraint obtained

by expressing that the square of the BRS operator δBRS vanishes (it has the

property of a cohomology operator):

2

δBRS = 0

and called the Wess“Zumino consistency conditions.

To calculate the BRS variation of AC, we need also the BRS transformation

of the fermion ghost C(x):

δBRS C(x) = µ C2 (x).

¯ (68)

216 J. Zinn-Justin

The condition that AC is BRS invariant,

d4 x A (Aµ ; x) C(x) = 0 ,

δBRS

yields a constraint on the possible form of anomalies that determines the term

cubic in A in the r.h.s. of (65) completely. One can verify that

d4 x tr C(x)‚µ Aν ‚ρ Aσ + 1 Aν Aρ Aσ

δBRS = 0.

µνρσ 2

Explicitly, after integration by parts, the equation takes the form

d4 x ‚µ C2 (x)Aν ‚ρ Aσ + ‚µ CDν C‚ρ Aσ + ‚µ CAν ‚ρ Dσ C

tr

µνρσ

+ 1 ‚µ C2 (x)Aν Aρ Aσ + 1 ‚µ C (Dν CAρ Aσ + Aν Dρ CAσ + Aν Aρ Dσ C) = 0 .

2 2

The terms linear in A, after integrating by parts the ¬rst term and using the

antisymmetry of the symbol, cancels automatically:

d4 x (‚µ C‚ν C‚ρ Aσ + ‚µ CAν ‚ρ ‚σ C) = 0 .

tr

µνρσ

In the same way, the cubic terms cancel (the anticommuting properties of C

have to be used):

d4 x {(‚µ CC + C‚µ C) Aν Aρ Aσ + ‚µ C ([Aν , C]CAρ Aσ

tr

µνρσ

+Aν [Aρ , C]Aσ + Aν Aρ [Aσ , C])} = 0 .

It is only the quadratic terms that give a relation between the quadratic and

cubic terms in the anomaly, both contributions being proportional to

d4 x ‚µ C‚ν CAρ Aσ .

tr

µνρσ

7 Lattice Fermions: Ginsparg“Wilson Relation

Notation. We now return to the problem of lattice fermions discussed in

Sect. 3.4. For convenience we set the lattice spacing a = 1 and use for the

¬elds the notation ψ(x) ≡ ψx .

Ginsparg“Wilson Relation. It had been noted, many years ago, that a po-

tential way to avoid the doubling problem while still retaining chiral properties

in the continuum limit was to look for lattice Dirac operators D that, instead of

anticommuting with γ5 , would satisfy the relation

D’1 γ5 + γ5 D’1 = γ5 1 (69)

Chiral Anomalies and Topology 217

where 1 stands for the identity both for lattice sites and in the algebra of γ-

matrices. More explicitly,

(D’1 )xy γ5 + γ5 (D’1 )xy = γ5 δxy .

More generally, the r.h.s. can be replaced by any local positive operator on

the lattice: locality of a lattice operator is de¬ned by a decrease of its matrix

elements that is at least exponential when the points x, y are separated. The

anti-commutator being local, it is expected that it does not a¬ect correlation

functions at large distance and that chiral properties are recovered in the con-

tinuum limit. Note that when D is the Dirac operator in a gauge background,

the condition (69) is gauge invariant.

However, lattice Dirac operators solutions to the Ginsparg“Wilson relation

(69) have only recently been discovered because the demands that both D and

the anticommutator {D’1 , γ5 } should be local seemed di¬cult to satisfy, spe-

cially in the most interesting case of gauge theories.

Note that while relation (69) implies some generalized form of chirality on

the lattice, as we now show, it does not guarantee the absence of doublers, as

examples illustrate. But the important point is that in this class solutions can

be found without doublers.

7.1 Chiral Symmetry and Index

We ¬rst discuss the main properties of a Dirac operator satisfying relation (69)

and then exhibit a generalized form of chiral transformations on the lattice.

Using the relation, quite generally true for any euclidean Dirac operator

satisfying hermiticity and re¬‚ection symmetry (see textbooks on symmetries of

euclidean fermions),

D† = γ5 Dγ5 , (70)

one can rewrite relation (69), after multiplication by γ5 , as

†

D’1 + D’1 =1

and, therefore,

D + D† = DD† = D† D . (71)

This implies that the lattice operator D has an index and, in addition, that

S=1’D (72)

is unitary:

SS† = 1 .

The eigenvalues of S lie on the unit circle. The eigenvalue one corresponds to

the pole of the Dirac propagator.

Note also the relations

γ5 S = S† γ5 , (γ5 S)2 = 1 . (73)

218 J. Zinn-Justin

The matrix γ5 S is hermitian and 1 (1 ± γ5 S) are two orthogonal projectors. If

2

D is a Dirac operator in a gauge background, these projectors depend on the

gauge ¬eld.

It is then possible to construct lattice actions that have a chiral symmetry

that corresponds to local but non point-like transformations. In the abelian

example,

eiθγ5 S xy ψy , ψx = ψx eiθγ5 .

¯ ¯

ψx = (74)

y

¯

(The reader is reminded that in the formalism of functional integrals, ψ and ψ are

independent integration variables and, thus, can be transformed independently.)

¯

Indeed, the invariance of the lattice action S(ψ, ψ),

¯ ¯ ¯

S(ψ, ψ) = ψx Dxy ψy = S(ψ , ψ ),

x,y

is implied by

eiθγ5 Deiθγ5 S = D ” Deiθγ5 S = e’iθγ5 D .

Using the second relation in (73), we expand the exponentials and reduce the

equation to

Dγ5 S = ’γ5 D , (75)

which is another form of relation (69).

However, the transformations (74), no longer leave the integration measure

of the fermion ¬elds,

¯

dψx dψx ,

x

automatically invariant. The jacobian of the change of variables ψ ’ ψ is

J = det eiθγ5 eiθγ5 S = det eiθγ5 (2’D) = 1 + iθ tr γ5 (2 ’ D) + O(θ2 ), (76)

where trace means trace in the space of γ matrices and in the lattice indices.

This leaves open the possibility of generating the expected anomalies, when the

Dirac operator of the free theory is replaced by the covariant operator in the

background of a gauge ¬eld, as we now show.

Eigenvalues of the Dirac Operator in a Gauge Background. We brie¬‚y