Anomalies

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 $ℝ^{4}$ with nonzero integral of

16 π^{2} k : = \int_{ℝ^{4}} t r F_{μ ν} {\tilde{F}}^{μ ν} .

(3.2.1)

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

exp [i \frac{𝜃}{16 π^{2}} t r F_{μ ν} {\tilde{F}}^{μ ν}] .

(3.2.2)

Therefore, a conﬁguration with the instanton number $k$ has a nontrivial phase $e^{i 𝜃 k}$ . Note that a shift of $𝜃$ by $2 π$ does not change this phase at all. Therefore, even in a quantum theory, the shift $𝜃 \to 𝜃 + 2 π$ is a symmetry.

Using

t r F_{μ ν} F_{μ ν} = \frac{1}{2} t r {(F_{μ ν} \pm {\tilde{F}}_{μ ν})}^{2} \mp t r F_{μ ν} {\tilde{F}}_{μ ν} \geq \mp t r F_{μ ν} {\tilde{F}}_{μ ν},

(3.2.3)

we ﬁnd that

\int d^{4} x t r F_{μ ν} F_{μ ν} \geq 16 π^{2} | k |

(3.2.4)

which is saturated only when

F_{μ ν} + {\tilde{F}}_{μ ν} \propto F_{α β} = 0 or F_{μ ν} - {\tilde{F}}_{μ ν} \propto F_{\dot{α} \dot{β}} = 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 [- \frac{1}{2 g^{2}} \int t r F_{μ ν} F^{μ ν} + i \frac{𝜃}{16 π^{2}} \int t r F_{μ ν} {\tilde{F}}^{μ ν}] = e^{2 π i τ k} .

(3.2.6)

We similarly have the contribution $e^{2 π i \bar{τ} | k |}$ in an anti-instanton background. The fact that we have just $τ$ or $\bar{τ}$ , instead of more complicated combinations, is related to the fact that in the instanton background in a supersymmetric theory, $δ λ_{\dot{α}} = F_{\dot{α} \dot{β}} 𝜖^{\dot{β}} = 0$ 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 $ψ_{α}$ coupled to the gauge ﬁeld, with the kinetic term

{\bar{ψ}}_{\dot{α}} D_{μ} σ^{μ \dot{α} α} ψ_{α} .

(3.2.7)

Let us say $ψ_{α}$ is in the representation $R$ of the gauge group. It is known that the number of zero modes in $ψ_{α}$ minus the number of zero modes in ${\bar{ψ}}_{\dot{α}}$ is $2 C (R) k$ . 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,

⟨ O_{1} O_{2} \dots ⟩ = \int [D ψ] [D \bar{ψ}] O_{1} O_{2} \dots e^{- S} = 0

(3.2.8)

unless the product of the operators $O_{1} O_{2} \dots$ contains $2 C (R) k$ more $ψ$ ’s than $\bar{ψ}$ ’s. This is interpreted as follows: the path integral measures $[D ψ]$ and $[D \bar{ψ}]$ contain both inﬁnite number of integrations. However, there is a ﬁnite number, $2 C (R) k$ , of diﬀerence in the number of integration variables. Equivalently, under the constant rotation

ψ \to e^{i φ} ψ, \bar{ψ} \to e^{- i φ} \bar{ψ},

(3.2.9)

the fermionic path integration measure rotates as

\begin{aligned} [D ψ] & \to [D ψ] e^{+ \infty i φ + 2 C (R) k i φ}, \\ [D \bar{ψ}] & \to [D \bar{ψ}] e^{- \infty i φ} . \end{aligned}

(3.2.10)

When combined, we have

[D ψ] [D \bar{ψ}] \to [D ψ] [D \bar{ψ}] e^{2 C (R) k i φ} = [D ψ] [D \bar{ψ}] exp [2 C (R) φ \frac{i}{16 π^{2}} \int t r F_{μ ν} {\tilde{F}}^{μ ν}] .

(3.2.11)

This can be compensated by a shift of the $𝜃$ angle, $𝜃 \to 𝜃 + 2 C (R) φ$ . As we recalled before, the shift $𝜃 \to 𝜃 + 2 π$ is a symmetry. Therefore, the rotation of the ﬁeld $ψ$ by $exp (\frac{2 π i}{2 C (R)})$ 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 $\bar{R}$ , 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 $S U (2)$ is anomalous, due to the following fact. When we perform the path integral of this system, we ﬁrst need to consider

Z [A_{μ}] = \int [D ψ_{α i}] [D {\bar{ψ}}_{\dot{α} i}] e^{- \int \bar{ψ} D_{μ} σ^{μ} ψ}

(3.2.12)

where $i = 1, 2$ is the $S U (2)$ doublet index. To perform a further integration over $A_{μ}$ consistently, we need

Z [A_{μ}] = Z [A_{μ}^{g}], A_{μ}^{g} = g^{- 1} A_{μ} g + g^{- 1} \partial_{μ} g .

(3.2.13)

for any gauge transformation $g : ℝ^{4} \to S U (2)$ . These maps are characterized by $π_{4} (S U (2))$ . It is known that

π_{4} (S U (2)) = π_{4} (S^{3}) = ℤ_{2} .

(3.2.14)

Let $g_{0} : ℝ^{4} \to S U (2)$ be the one corresponding to the nontrivial element in this $ℤ_{2}$ . Then it is known that

[D ψ_{α i}] [D {\bar{ψ}}_{\dot{α} i}] \overset{g_{0}}{\to} - [D ψ_{α i}] [D {\bar{ψ}}_{\dot{α} i}]

(3.2.15)

resulting in

Z [A_{μ}^{g_{0}}] = - Z [A_{μ}],

(3.2.16)

thus making the path integral over $A_{μ}$ inconsistent.

In general $π_{4} (G) = ℤ_{2}$ if $G = S p (n)$ , and $π_{4} (G) = 1$ otherwise. Therefore Witten’s global anomaly can be there only for Weyl fermions in a representation $R$ under gauge $S p (n)$ . A short computation reveals that there is an anomaly only when $C (R)$ is half-integral.

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

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