3.2 Anomalies

3.2.1 Anomalies of global symmetry

Non-abelian gauge theories have an important source of non-perturbative effects, called instantons. This is a nontrivial classical field configuration in the Euclidean 4 with nonzero integral of

16π2k :=4trFμνF̃μν. (3.2.1)

In the standard normalization of the trace for SU(N), k is automatically an integer, and is called the instanton number. The theta term in the Euclidean path integral appears as

exp i 𝜃 16π2trFμνF̃μν. (3.2.2)

Therefore, a configuration with the instanton number k has a nontrivial phase ei𝜃k. Note that a shift of 𝜃 by 2π does not change this phase at all. Therefore, even in a quantum theory, the shift 𝜃 𝜃 + 2π is a symmetry.


trFμνFμν = 1 2tr(Fμν ±F̃μν)2 trFμνF̃μν trFμνF̃μν, (3.2.3)

we find that

d4xtrFμνFμν 16π2|k| (3.2.4)

which is saturated only when

Fμν + F̃μν Fαβ = 0orFμν F̃μν Fα̇β̇ = 0 (3.2.5)

depending on the sign of k. Therefore, within configurations of fixed k, those satisfying relations (3.2.5) give the dominant contributions to the path integral. The solutions to (3.2.5) are called instantons or anti-instantons, depending on the sign of k.

In an instanton background, the weight in the path integral coming from the gauge kinetic term is

exp 1 2g2trFμνFμν + i 𝜃 16π2trFμνF̃μν = e2πiτk. (3.2.6)

We similarly have the contribution e2πiτ̄|k| in an anti-instanton background. The fact that we have just τ or τ̄, instead of more complicated combinations, is related to the fact that in the instanton background in a supersymmetric theory, δλα̇ = Fα̇β̇𝜖β̇ = 0 assuming the D-term is also zero, and thus the dotted supertranslation is preserved. Similarly, the undotted supersymmetry is unbroken in the anti-instanton background.

Now, consider charged Weyl fermions ψα coupled to the gauge field, with the kinetic term

ψ̄α̇Dμσμα̇αψα. (3.2.7)

Let us say ψα is in the representation R of the gauge group. It is known that the number of zero modes in ψα minus the number of zero modes in ψ̄α̇ is 2C(R)k. In particular, the path integral restricted to the k-instanton configuration with positive k is vanishing unless we insert k additional ψ’s in the integrand. More explicitly,

O1O2 =[Dψ][Dψ̄]O1O2eS = 0 (3.2.8)

unless the product of the operators O1O2 contains 2C(R)k more ψ’s than ψ̄’s. This is interpreted as follows: the path integral measures [Dψ] and [Dψ̄] contain both infinite number of integrations. However, there is a finite number, 2C(R)k, of difference in the number of integration variables. Equivalently, under the constant rotation

ψ eiφψ,ψ̄ eiφψ̄, (3.2.9)

the fermionic path integration measure rotates as

[Dψ] [Dψ]e+iφ + 2C(R)kiφ, [Dψ̄] [Dψ̄]eiφ. (3.2.10)

When combined, we have

[Dψ][Dψ̄] [Dψ][Dψ̄]e2C(R)kiφ = [Dψ][Dψ̄] exp 2C(R)φ i 16π2trFμνF̃μν. (3.2.11)

This can be compensated by a shift of the 𝜃 angle, 𝜃 𝜃 + 2C(R)φ. As we recalled before, the shift 𝜃 𝜃 + 2π is a symmetry. Therefore, the rotation of the field ψ by exp( 2πi 2C(R)) is a genuine, unbroken symmetry.

3.2.2 Anomalies of gauge symmetry

In 𝒩=2 gauge theories, fermions always come in non-chiral representations. Indeed, the fermions in the vector multiplets are always in the adjoint, the 𝒩=1 chiral superfields in a full hypermultiplet is a sum of a representation R and its conjugate R̄, and a half-hypermultiplet counts as an 𝒩=1 chiral superfield in a pseudo-real representation R. Therefore there are no perturbative gauge anomalies.

One needs to be careful about Witten’s global anomaly [38], though, as this can arise even for real representations. It is known that a Weyl fermion in the doublet of gauge SU(2) is anomalous, due to the following fact. When we perform the path integral of this system, we first need to consider

Z[Aμ] =[Dψαi][Dψ̄α̇i]eψ̄Dμσμψ (3.2.12)

where i = 1, 2 is the SU(2) doublet index. To perform a further integration over Aμ consistently, we need

Z[Aμ] = Z[Aμg],Aμg = g1Aμg + g1μg. (3.2.13)

for any gauge transformation g : 4 SU(2). These maps are characterized by π4(SU(2)). It is known that

π4(SU(2)) = π4(S3) = 2. (3.2.14)

Let g0 : 4 SU(2) be the one corresponding to the nontrivial element in this 2. Then it is known that

[Dψαi][Dψ̄α̇i]g0 [Dψαi][Dψ̄α̇i] (3.2.15)

resulting in

Z[Aμg0] = Z[Aμ], (3.2.16)

thus making the path integral over Aμ inconsistent.

In general π4(G) = 2 if G = Sp(n), and π4(G) = 1 otherwise. Therefore Witten’s global anomaly can be there only for Weyl fermions in a representation R under gauge Sp(n). A short computation reveals that there is an anomaly only when C(R) is half-integral.

Witten’s anomaly is always 2 valued in four dimensions. Therefore full hypermultiplets are always free of Witten’s global anomaly. The danger only exists for half-hypermultiplets of gauge Sp(n). For example, one cannot have odd number of half-hypermultiplets in the doublet representation of gauge SU(2), or more generally, one cannot have half-hypermultiplets in a pseudo-real representation R of gauge Sp(n) such that C(R) is half-integral.