background. We assume that its spectrum is discrete (this is certainly true on

a ¬nite lattice where D is a matrix). The operator D is related by (72) to a

unitary operator S whose eigenvalues have modulus one. Therefore, if we denote

by |n its nth eigenvector,

’ D† |n = (1 ’ e’iθn ) |n .

D |n = (1 ’ S) |n = (1 ’ eiθn ) |n

Then, using (70), we infer

Dγ5 |n = (1 ’ e’iθn )γ5 |n .

Chiral Anomalies and Topology 219

The discussion that follows then is analogous to the discussion of Sect. 4.5 to

which we refer for details. We note that when the eigenvalues are not real, θn = 0

(mod π), γ5 |n is an eigenvector di¬erent from |n because the eigenvalues are

di¬erent. Instead, in the two subspaces corresponding to the eigenvalues 0 and

2, we can choose eigenvectors with de¬nite chirality

γ5 |n = ± |n .

We call below n± the number of eigenvalues 0, and ν± the number of eigenvalues

2 with chirality ±1.

Note that on a ¬nite lattice δxy is a ¬nite matrix and, thus,

tr γ5 δxy = 0 .

Therefore,

tr γ5 (2 ’ D) = ’ tr γ5 D ,

which implies

n| γ5 (2 ’ D) |n = ’ n| γ5 D |n .

n n

In the equation all complex eigenvalues cancel because the vectors |n and γ5 |n

are orthogonal. The sum reduces to the subspace of real eigenvalues, where

the eigenvectors have de¬nite chirality. On the l.h.s. only the eigenvalue 0 con-

tributes, and on the r.h.s. only the eigenvalue 2. We ¬nd

n+ ’ n’ = ’(ν+ ’ ν’ ).

This equation tells us that the di¬erence between the number of states of di¬erent

chirality in the zero eigenvalue sector is cancelled by the di¬erence in the sector

of eigenvalue two (which corresponds to very massive states).

Remark. It is interesting to note the relation between the spectrum of D and

the spectrum of γ5 D, which from relation (70) is a hermitian matrix,

γ5 D = D† γ5 = (γ5 D)† ,

and, thus, diagonalizable with real eigenvalues. It is simple to verify the following

two equations, of which the second one is obtained by changing θ into θ + 2π,

γ5 D(1 ’ ieiθn /2 γ5 ) |n = 2 sin(θn /2)(1 ’ ieiθn /2 γ5 ) |n ,

γ5 D(1 + ieiθn /2 γ5 ) |n = ’2 sin(θn /2)(1 + ieiθn /2 γ5 ) |n .

These equations imply that the eigenvalues ±2 sin(θn /2) of γ5 D are paired except

for θn = 0 (mod π) where |n and γ5 |n are proportional. For θn = 0, γ5 D has

also eigenvalue 0. For θn = π, γ5 D has eigenvalue ±2 depending on the chirality

of |n .

In the same way,

γ5 (2 ’ D)(1 + eiθn /2 γ5 ) |n = 2 cos(θn /2)(1 + eiθn /2 γ5 ) |n ,

γ5 (2 ’ D)(1 ’ eiθn /2 γ5 ) |n = ’2 cos(θn /2)(1 ’ eiθn /2 γ5 ) |n .

220 J. Zinn-Justin

Jacobian and Lattice Anomaly. The variation of the jacobian (76) can now

be evaluated. Opposite eigenvalues of γ5 (2 ’ D) cancel. The eigenvalues for

θn = π give factors one. Only θn = 0 gives a non-trivial contribution:

J = det eiθγ5 (2’D) = e2iθ(n+ ’n’ ) .

The quantity tr γ5 (2 ’ D), coe¬cient of the term of order θ, is a sum of terms

that are local, gauge invariant, pseudoscalar, and topological as the continuum

anomaly (36) since

tr γ5 (2 ’ D) = n| γ5 (2 ’ D) |n = 2(n+ ’ n’ ).

n

Non-Abelian Generalization. We now consider the non-abelian chiral trans-

formations

+ γ5 S)U+ + 1 (1 ’ γ5 S)U’ ψ ,

1

ψU = 2 (1 2

+ γ5 )U† + 1 (1 ’ γ5 )U† ,

¯ ¯ 1

ψU = ψ 2 (1 (77)

’ +

2

where U± are matrices belonging to some unitary group G. Near the identity

U = 1 + ˜ + O(˜2 ),

where ˜ is an element of the Lie algebra.

¯

We note that this amounts to de¬ne di¬erently chiral components of ψ and

ψ, for ψ the de¬nition being even gauge ¬eld dependent.

We assume that G is a vector symmetry of the fermion action, and thus the

Dirac operator commutes with all elements of the Lie algebra:

[D, ˜] = 0 .

Then, again, the relation (69) in the form (75) implies the invariance of the

fermion action:

¯ ¯

ψU D ψ U = ψ D ψ .

The jacobian of an in¬nitesimal chiral transformation ˜ = ˜+ = ’˜’ is

J = 1 + tr γ5 ˜(2 ’ D) + O(˜2 ).

Wess“Zumino Consistency Conditions. To determine anomalies in the case

of gauge ¬elds coupling di¬erently to fermion chiral components, one can on the

lattice also play with the property that BRS transformations are nilpotent. They

take the form

δUxy = µ (Cx Uxy ’ Uxy Cy ) ,

¯

δCx = µC2 ,

¯x

instead of (66), (68). Moreover, the matrix elements Dxy of the gauge covariant

Dirac operator transform like Uxy .

Chiral Anomalies and Topology 221

7.2 Explicit Construction: Overlap Fermions

An explicit solution of the Ginsparg“Wilson relation without doublers can be

derived from operators DW that share the properties of the Wilson“Dirac opera-

tor of (17), that is which avoid doublers at the price of breaking chiral symmetry

explicitly. Setting

A = 1 ’ DW /M , (78)

where M > 0 is a mass parameter that must chosen, in particular, such that A

has no zero eigenvalue, one takes

’1/2 ’1/2

S = A A† A ’ D = 1 ’ A A† A . (79)

The matrix A is such that

A† = γ5 Aγ5 ’ B = γ5 A = B† .

The hermitian matrix B has real eigenvalues. Moreover,

1/2

B† B = B 2 = A † A ’ A† A = |B|.

We conclude

γ5 S = sgn B ,

where sgn B is the matrix with the same eigenvectors as B, but all eigenvalues

replaced by their sign. In particular this shows that (γ5 S)2 = 1.

’1/2

With this ansatz D has a zero eigenmode when A A† A has the eigen-

†

value one. This can happen when A and A have the same eigenvector with a

positive eigenvalue.

This is the idea of overlap fermions, the name overlap refering only to the

way this Dirac operator was initially introduced.

Free Fermions. We now verify the absence of doublers for vanishing gauge

¬elds. The Fourier representation of a Wilson“Dirac operator has the general

form

DW (p) = ±(p) + iγµ βµ (p), (80)

where ±(p) and βµ (p) are real, periodic, smooth functions. In the continuum

limit, one must recover the usual massless Dirac operator, which implies

βµ (p) ∼ pµ , ±(p) ≥ 0 , ±(p) = O(p2 ),

|p|’0 |p|’0

and ±(p) > 0 for all values of pµ such that βµ (p) = 0 for |p| = 0 (i.e. all values

that correspond to doublers). Equation (17) in the limit m = 0 provides an

explicit example.

Doublers appear if the determinant of the overlap operator D (78, 79) van-

ishes for |p| = 0. In the example of the operator (80), a short calculation shows

that this happens when

2

2

M ’ ±(p) ’ M + ±(p) 2

2

+ βµ (p) + βµ (p) = 0 .

222 J. Zinn-Justin

This implies βµ (p) = 0, an equation that necessarily admits doubler solutions,

and

|M ’ ±(p)| = M ’ ±(p).

The solutions to this equation depend on the value of ±(p) with respect to M for

the doubler modes, that is for the values of p such that βµ (p) = 0. If ±(p) ¤ M

the equation is automatically satis¬ed and the corresponding doubler survives.

As mentioned in the introduction to this section, the relation (69) alone does not

guarantee the absence of doublers. Instead, if ±(p) > M , the equation implies

±(p) = M , which is impossible. Therefore, by rescaling ±(p), if necessary, we

can keep the wanted pµ = 0 mode while eliminating all doublers. The modes

associated to doublers for ±(p) ¤ M then, instead, correspond to the eigenvalue

2 for D, and the doubling problem is solved, at least in a free theory.

In presence of a gauge ¬eld, the argument can be generalized provided the

plaquette terms in the lattice action are constrained to remain su¬ciently close

to one.

Remark. Let us stress that, if it seems that the doubling problem has been

solved from the formal point of view, from the numerical point of view the

calculation of the operator (A† A)’1/2 in a gauge background represents a major

challenge.

8 Supersymmetric Quantum Mechanics

and Domain Wall Fermions

Because the construction of lattice fermions without doublers we have just de-

scribed is somewhat arti¬cial, one may wonder whether there is a context in

which they would appear more naturally. Therefore, we now brie¬‚y outline

how a similar lattice Dirac operator can be generated by embedding ¬rst four-

dimensional space in a larger ¬ve-dimensional space. This is the method of do-

main wall fermions.

Because the general idea behind domain wall fermions has emerged ¬rst in

another context, as a preparation, we ¬rst recall a few properties of the spectrum

of the hamiltonian in supersymmetric quantum mechanics, a topic also related

to the index of the Dirac operator (Sect. 4.5), and very directly to stochastic

dynamics in the form of Langevin or Fokker“Planck equations.

8.1 Supersymmetric Quantum Mechanics

We now construct a quantum theory that exhibits the simplest form of super-

symmetry where space“time reduces to time only. We know that this reduces

¬elds to paths and, correspondingly, quantum ¬eld theory to simple quantum

mechanics.

We ¬rst introduce a ¬rst order di¬erential operator D acting on functions of

one real variable, which is a 2 — 2 matrix (σi still are the Pauli matrices):

D ≡ σ1 dx ’ iσ2 A(x) (81)

Chiral Anomalies and Topology 223

(dx ≡ d/dx). The function A(x) is real and, thus, the operator D is anti-

hermitian.

The operator D shares several properties with the Dirac operator of Sect. 4.5.

In particular, it satis¬es

σ3 D + Dσ3 = 0 ,

and, thus, has an index (σ3 playing the role of γ5 ). We introduce the operator

D = dx + A(x) ’ D† = ’dx + A(x),

and

00 01

’ Q† = D†

Q=D .

10 00

Then,

D = Q ’ Q† ,

Q2 = (Q† )2 = 0 . (82)

We consider now the positive semi-de¬nite hamiltonian, anticommutator of Q

and Q† ,

D† D 0

H = QQ† + Q† Q = ’D2 = .

DD†

0

The relations (82) imply that

[H, Q] = [H, Q† ] = 0 .

The operators Q, Q† are the generators of the simplest form of a supersymmetric

algebra and the hamiltonian H is supersymmetric.

The eigenvectors of H have the form ψ+ (x)(1, 0) and ψ’ (x)(0, 1) and satisfy,

respectively,

D† D |ψ+ = µ+ |ψ+ , DD† |ψ’ = µ’ |ψ’ , µ± ≥ 0 ,

and (83)

where

D† D = ’d2 + A2 (x) ’ A (x), DD† = ’d2 + A2 (x) + A (x).

x x

Moreover, if x belongs to a bounded interval or A(x) ’ ∞ for |x| ’ ∞, then

the spectrum of H is discrete.

Multiplying the ¬rst equation in (83) by D, we conclude that if D |ψ+ = 0

and, thus, + does not vanish, it is an eigenvector of DD† with eigenvalue µ+ ,

and conversely. Therefore, except for a possible ground state with vanishing

eigenvalue, the spectrum of H is doubly degenerate.

This observation is consistent with the analysis of Sect. 4.5 applied to the

operator D. We know from that section that either eigenvectors are paired

√

|ψ , σ3 |ψ with opposite eigenvalues ±i µ, or they correspond to the eigenvalue

zero and can be chosen with de¬nite chirality

D |ψ = 0 , σ3 |ψ = ± |ψ .

224 J. Zinn-Justin

It is convenient to now introduce the function S(x):

S (x) = A(x),

and for simplicity discuss only the situation of operators on the entire real line.

We assume that

S(x)/|x| ≥ > 0.

x’±∞

Then the function S(x) is such that e’S(x) is a normalizable wave function:

dx e’2S(x) < ∞.

In the stochastic interpretation where D† D has the interpretation of a Fok-

ker“Planck hamiltonian generating the time evolution of some probability dis-

tribution, e’2S(x) is the equilibrium distribution.

When e’S(x) is normalizable, we know one eigenvector with vanishing eigen-