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

$$16{\pi}^{2}k:={\int}_{{\mathbb{R}}^{4}}tr\phantom{\rule{0.3em}{0ex}}{F}_{\mu \nu}{\stackrel{\u0303}{F}}^{\mu \nu}.$$ | (3.2.1) |

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

$$exp\left[i\frac{\mathit{\theta}}{16{\pi}^{2}}tr\phantom{\rule{0.3em}{0ex}}{F}_{\mu \nu}{\stackrel{\u0303}{F}}^{\mu \nu}\right].$$ | (3.2.2) |

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

Using

we ﬁnd that

$$\int {d}^{4}xtr\phantom{\rule{0.3em}{0ex}}{F}_{\mu \nu}{F}_{\mu \nu}\ge 16{\pi}^{2}\left|k\right|$$ | (3.2.4) |

which is saturated only when

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

We similarly have the contribution ${e}^{2\pi i\stackrel{\u0304}{\tau}\left|k\right|}$ in an anti-instanton background. The fact that we have just $\tau $ or $\stackrel{\u0304}{\tau}$, instead of more complicated combinations, is related to the fact that in the instanton background in a supersymmetric theory, $\delta {\lambda}_{\stackrel{\u0307}{\alpha}}={F}_{\stackrel{\u0307}{\alpha}\stackrel{\u0307}{\beta}}{\mathit{\epsilon}}^{\stackrel{\u0307}{\beta}}=0$ 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 ${\psi}_{\alpha}$ coupled to the gauge ﬁeld, with the kinetic term

$${\stackrel{\u0304}{\psi}}_{\stackrel{\u0307}{\alpha}}D{\phantom{\rule{0.0pt}{0ex}}}_{\mu}{\sigma}^{\mu \stackrel{\u0307}{\alpha}\alpha}{\psi}_{\alpha}.$$ | (3.2.7) |

Let us say ${\psi}_{\alpha}$ is in the representation $R$ of the gauge group. It is known that the number of zero modes in ${\psi}_{\alpha}$ minus the number of zero modes in ${\stackrel{\u0304}{\psi}}_{\stackrel{\u0307}{\alpha}}$ is $2C\left(R\right)k$. In particular, the path integral restricted to the $k$-instanton conﬁguration with positive $k$ is vanishing unless we insert $k$ additional $\psi $’s in the integrand. More explicitly,

$$\u27e8{O}_{1}{O}_{2}\cdots \phantom{\rule{0.3em}{0ex}}\u27e9=\int \left[D\psi \right]\left[D\stackrel{\u0304}{\psi}\right]{O}_{1}{O}_{2}\cdots {e}^{-S}=0$$ | (3.2.8) |

unless the product of the operators ${O}_{1}{O}_{2}\cdots \phantom{\rule{0.3em}{0ex}}$ contains $2C\left(R\right)k$ more $\psi $’s than $\stackrel{\u0304}{\psi}$’s. This is interpreted as follows: the path integral measures $\left[D\psi \right]$ and $\left[D\stackrel{\u0304}{\psi}\right]$ contain both inﬁnite number of integrations. However, there is a ﬁnite number, $2C\left(R\right)k$, of diﬀerence in the number of integration variables. Equivalently, under the constant rotation

$$\psi \to {e}^{i\phi}\psi ,\phantom{\rule{1em}{0ex}}\stackrel{\u0304}{\psi}\to {e}^{-i\phi}\stackrel{\u0304}{\psi},$$ | (3.2.9) |

the fermionic path integration measure rotates as

When combined, we have

This can be compensated by a shift of the $\mathit{\theta}$ angle, $\mathit{\theta}\to \mathit{\theta}+2C\left(R\right)\phi $. As we recalled before, the shift $\mathit{\theta}\to \mathit{\theta}+2\pi $ is a symmetry. Therefore, the rotation of the ﬁeld $\psi $ by $exp\left(\frac{2\pi i}{2C\left(R\right)}\right)$ is a genuine, unbroken symmetry.

In $\mathcal{\mathcal{N}}=2$ gauge theories, fermions always come in non-chiral representations. Indeed, the fermions in the vector multiplets are always in the adjoint, the $\mathcal{\mathcal{N}}=1$ chiral superﬁelds in a full hypermultiplet is a sum of a representation $R$ and its conjugate $\stackrel{\u0304}{R}$, and a half-hypermultiplet counts as an $\mathcal{\mathcal{N}}=1$ 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 $SU\left(2\right)$ is anomalous, due to the following fact. When we perform the path integral of this system, we ﬁrst need to consider

where $i=1,2$ is the $SU\left(2\right)$ doublet index. To perform a further integration over ${A}_{\mu}$ consistently, we need

$$Z\left[{A}_{\mu}\right]=Z\left[{A}_{\mu}^{g}\right],\phantom{\rule{1em}{0ex}}{A}_{\mu}^{g}={g}^{-1}{A}_{\mu}g+{g}^{-1}{\partial}_{\mu}g.$$ | (3.2.13) |

for any gauge transformation $g:{\mathbb{R}}^{4}\to SU\left(2\right)$. These maps are characterized by ${\pi}_{4}\left(SU\left(2\right)\right)$. It is known that

$${\pi}_{4}\left(SU\left(2\right)\right)={\pi}_{4}\left({S}^{3}\right)={\mathbb{Z}}_{2}.$$ | (3.2.14) |

Let ${g}_{0}:{\mathbb{R}}^{4}\to SU\left(2\right)$ be the one corresponding to the nontrivial element in this ${\mathbb{Z}}_{2}$. Then it is known that

resulting in

$$Z\left[{A}_{\mu}^{{g}_{0}}\right]=-Z\left[{A}_{\mu}\right],$$ | (3.2.16) |

thus making the path integral over ${A}_{\mu}$ inconsistent.

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

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