granted and continue our argument. First, we de¬ne a generalized connection on

the bundle by

1 1

A(ξ) ≡ ’ a’1 (ξ) da(ξ) + a’1 (ξ)A(x)a(ξ) = ’ C(ξ) + A(ξ). (393)

g g

It follows that the corresponding ¬eld strength on the bundle is

2

1 1

F = dA ’ gA = (d + s) A ’ C ’g A’ C

2

g g (394)

= dA ’ gA2 = F.

To go from the ¬rst to the second line we have used (380). This result is some-

times referred to as the Russian formula [43]. The result implies that the com-

ponents of the generalized ¬eld strength in the directions of the group manifold

all vanish.

Aspects of BRST Quantization 163

It is now obvious that

Chn [A] = Tr Fn = Chn [A]; (395)

moreover, F satis¬es the Bianchi identity

DF = dF ’ g[A, F] = 0. (396)

Again, this leads us to infer that

’ 0 0

dChn [A] = 0 Chn [A] = dω2n’1 [A] = dω2n’1 [A], (397)

where the last equality follows from (395) and (392). The middle step, which

states that the (2n ’ 1)-form of which Chn [A] is the total exterior derivative has

the same functional form in terms of A, as the one of which it is the exterior

space-time derivative has in terms of A, will be justi¬ed shortly.

We ¬rst conclude the derivation of the chain of descent equations, which

follow from the last result by expansion in terms of C:

dω2n’1 [A] = (d + s) ω2n’1 [A ’ C/g]

0 0

11 1 2n’1

0

= (d + s) ω2n’1 [A] + ω2n’2 [A, C] + ... + 2n’1 ω0 [A, C] .

g g

(398)

Comparing terms of the same degree, we ¬nd

0 0

dω2n’1 [A] = dω2n’1 [A],

’gsω2n’1 [A] = dω2n’1 [A, C],

0 1

’gsω2n’2 [A, C] = dω2n’3 [A, C],

1 2

(399)

...

’gsω0

2n’1

[A, C] = 0.

0 0

Obviously, this result carries over to the BRST di¬erentials: with In [A] = ωn [A],

one obtains

m + k = 2n ’ 1, k = 0, 1, 2, ..., 2n ’ 1.

k+1

k

δ„¦ Im [A, c] = dIm’1 [A, c], (400)

The ¬rst line just states the gauge independence of the Chern character. Taking

n = 3, we ¬nd that the third line is the Wess“Zumino consistency condition

(386):

1 2

δ„¦ I4 [A, c] = dI3 [A, c].

164 J.W. van Holten

Proofs and Solutions. We now show how to derive the result (392); this will

provide us at the same time with the tools to solve the Wess“Zumino consistency

condition. Consider an arbitrary gauge ¬eld con¬guration described by the Lie-

algebra valued 1-form A. From this we de¬ne a whole family of gauge ¬elds

t ∈ [0, 1].

At = tA, (401)

It follows, that

Ft ≡ F [At ] = tdA ’ gt2 A2 = tF [A] ’ g(t2 ’ t)A2 . (402)

This ¬eld strength 2-form satis¬es the appropriate Bianchi identity:

Dt Ft = dFt ’ g[At , F ] = 0. (403)

In addition, one easily derives

dFt

= dA ’ [At , A]+ = Dt A, (404)

dt

where the anti-commutator of the 1-forms implies a commutator of the Lie-

algebra elements. Now we can compute the Chern character

1 1

d

dt Tr (Dt A)Ftn’1

dt TrFtn = n

Chn [A] =

dt

0 0

(405)

1

dt Tr AFtn’1 .

= nd

0

In this derivation we have used both (404) and the Bianchi identity (403).

0 1

It is now straightforward to compute the forms ω5 and ω4 . First, taking n = 3

0

in the result (405) gives Ch3 [A] = dω5 with

1

g2 5

g3

0 0

dt Tr (Dt A)Ft2 = Tr AF 2 +

I5 [A] = ω5 [A] =3 AF+ A . (406)

2 10

0

0 1

Next, using (383), the BRST di¬erential of this expression gives δ„¦ I5 = dI4 ,

with

g2 4

g

= ’Tr c F +

1 2

2 2

I4 [A, c] A F + AF A + F A + A . (407)

2 2

This expression determines the anomaly up to a constant of normalization N :

g2 4

g

“a [A] = N Tr Ta F 2 + A2 F + AF A + F A2 + A . (408)

2 2

Of course, the component form depends on the gauge group; for example, for

SU (2) SO(3) it vanishes identically, which is true for any orthogonal group

SO(N ); in contrast the anomaly does not vanish identically for SU (N ), for any

N ≥ 3. In that case it has to be anulled by cancellation between the contributions

of chiral fermions in di¬erent representations of the gauge group G.

Aspects of BRST Quantization 165

Appendix. Conventions

In these lecture notes the following conventions are used. Whenever two objects

carrying a same index are multiplied (as in ai bi or in uµ v µ ) the index is a dummy

index and is to be summed over its entire range, unless explicitly stated other-

wise (summation convention). Symmetrization of objects enclosed is denoted by

braces {...}, anti-symmetrization by square brackets [...]; the total weight of such

(anti-)symmetrizations is always unity.

In these notes we deal both with classical and quantum hamiltonian sys-

tems. To avoid confusion, we use braces { , } to denote classical Poisson brack-

ets, brackets [ , ] to denote commutators and su¬xed brackets [ , ]+ to denote

anti-commutators.

The Minkowski metric ·µν has signature (’1, +1, ..., +1), the ¬rst co-ordinate

in a pseudo-cartesian co-ordinate system x0 being time-like. Arrows above sym-

bols (x) denote purely spatial vectors (most often 3-dimensional).

Unless stated otherwise, we use natural units in which c = = 1. Therefore

we usually do not write these dimensional constants explicitly. However, in a

few places where their role as universal constants is not a priori obvious they are

included in the equations.

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Chiral Anomalies and Topology

J. Zinn-Justin

Dapnia, CEA/Saclay, D´partement de la Direction des Sciences de la Mati`re, and

e e

Institut de Math´matiques de Jussieu“Chevaleret, Universit´ de Paris VII

e e

Abstract. When a ¬eld theory has a symmetry, global or local like in gauge theories,

in the tree or classical approximation formal manipulations lead to believe that the

symmetry can also be implemented in the full quantum theory, provided one uses the

proper quantization rules. While this is often true, it is not a general property and,

therefore, requires a proof because simple formal manipulations ignore the unavoidable

divergences of perturbation theory. The existence of invariant regularizations allows

solving the problem in most cases but the combination of gauge symmetry and chi-

ral fermions leads to subtle issues. Depending on the speci¬c group and ¬eld content,

anomalies are found: obstructions to the quantization of chiral gauge symmetries. Be-

cause anomalies take the form of local polynomials in the ¬elds, are linked to local

group transformations, but vanish for global (rigid) transformations, one discovers that

they have a topological character. In these notes we review various perturbative and

non-perturbative regularization techniques, and show that they leave room for possible

anomalies when both gauge ¬elds and chiral fermions are present. We determine the