So far we mainly considered the weak coupling limit $q\to 0$. Instead, consider sending $q\to \infty $, as shown in Fig. 9.8. We immediately ﬁnd that the strong coupling limit $q\to \infty $ is the weak coupling limit ${q}^{\prime}=1\u2215q\to 0$ of a similarly-looking $SU\left(2\right)$ gauge theory with four ﬂavors. Note however that the role of the singularities $B$ and $C$ are exchanged.

Originally, we had four ﬂavors with masses

$$\pm {\mu}_{1},\phantom{\rule{1em}{0ex}}\pm {\mu}_{2},\phantom{\rule{1em}{0ex}}\pm {\mu}_{3},\phantom{\rule{1em}{0ex}}\pm {\mu}_{4}.$$ | (9.4.1) |

The residues of $\lambda $ at the punctures were then

$$\pm {\mu}_{A},\phantom{\rule{1em}{0ex}}\pm {\mu}_{B},\phantom{\rule{1em}{0ex}}\pm {\mu}_{C},\phantom{\rule{1em}{0ex}}\pm {\mu}_{D}$$ | (9.4.2) |

with

The original masses ${\mu}_{i}$ are

Now the singularities $B$ and $C$ are exchanged. Then, the masses ${\mu}_{i}^{\prime}$ of the four hypermultiplets of the theory with the coupling ${q}^{\prime}=1\u2215q$ are instead given by

$$\begin{array}{lll}\hfill {\mu}_{1}^{\prime}& ={\mu}_{A}+{\mu}_{C}=\frac{{\mu}_{1}-{\mu}_{2}}{2}+\frac{{\mu}_{3}+{\mu}_{4}}{2},\phantom{\rule{2em}{0ex}}& \hfill \text{(9.4.5)}\\ \hfill {\mu}_{2}^{\prime}& =-{\mu}_{A}+{\mu}_{C}=-\frac{{\mu}_{1}-{\mu}_{2}}{2}+\frac{{\mu}_{3}+{\mu}_{4}}{2},\phantom{\rule{2em}{0ex}}& \hfill \text{(9.4.6)}\\ \hfill {\mu}_{3}^{\prime}& ={\mu}_{B}+{\mu}_{D}=\frac{{\mu}_{1}+{\mu}_{2}}{2}+\frac{{\mu}_{3}-{\mu}_{4}}{2},\phantom{\rule{2em}{0ex}}& \hfill \text{(9.4.7)}\\ \hfill {\mu}_{4}^{\prime}& ={\mu}_{B}-{\mu}_{D}=\frac{{\mu}_{1}+{\mu}_{2}}{2}-\frac{{\mu}_{3}-{\mu}_{4}}{2}.\phantom{\rule{2em}{0ex}}& \hfill \text{(9.4.8)}\end{array}$$The original masses (9.4.1) can be thought of as the weights of the vector representation of $SO\left(8\right)$. The dual masses $\pm {\mu}_{i}^{\prime}$ are then the weights of the spinor representation of $SO\left(8\right)$.

The dual quarks, therefore, transform in the spinor representation of the ﬂavor $SO\left(8\right)$ symmetry. We can identify these dual quarks as the monopoles in the original description. This can be seen by slowly changing the value of $q$, following how various paths on the sphere change, see Fig. 9.9. Originally, the path connecting branch points close to the singularity $B$ and $D$ was a monopole. Recall also that the semiclassical quantization of the monopole gave us a multiplet in the spinor representation of the ﬂavor symmetry $SO\left(2{N}_{f}\right)$ as we saw in Sec. 1.3. In the limit $q\to \infty $, these monopoles become excitations whose paths are totally contained in the sphere on the right. They are now the quark hypermultiplets in the trifundamental, as shown in Fig. 9.7.

The same manipulation also shows that the $SU\left(2\right)$ W-bosons in the dual description came from monopoles in the original description, see Fig. 9.10. Therefore it is important to keep in mind that the dual $SU\left(2\right)$ gauge multiplet is not the same physical excitation as the original $SU\left(2\right)$ gauge multiplet. Note also that this monopole has twice the magnetic charge of the monopole which became the dual quarks.

There is also a limit where the singularity $B$ approaches the singularity $C$, $q\to 1$. This is again equivalent to a weakly-coupled $SU\left(2\right)$ gauge theory with four ﬂavors, but with the role of the singularities are permuted, see Fig. 9.11. The four mass parameters of the hypermultiplets are now given by

$$\begin{array}{lll}\hfill {\mu}_{1}^{\u2033}& ={\mu}_{A}+{\mu}_{D}=\frac{{\mu}_{1}-{\mu}_{2}}{2}+\frac{{\mu}_{3}-{\mu}_{4}}{2},\phantom{\rule{2em}{0ex}}& \hfill \text{(9.4.9)}\\ \hfill {\mu}_{2}^{\u2033}& =-{\mu}_{A}+{\mu}_{D}=-\frac{{\mu}_{1}-{\mu}_{2}}{2}+\frac{{\mu}_{3}-{\mu}_{4}}{2},\phantom{\rule{2em}{0ex}}& \hfill \text{(9.4.10)}\\ \hfill {\mu}_{3}^{\u2033}& ={\mu}_{C}+{\mu}_{B}=\frac{{\mu}_{3}+{\mu}_{4}}{2}+\frac{{\mu}_{1}+{\mu}_{2}}{2},\phantom{\rule{2em}{0ex}}& \hfill \text{(9.4.11)}\\ \hfill {\mu}_{4}^{\u2033}& ={\mu}_{C}-{\mu}_{B}=\frac{{\mu}_{3}+{\mu}_{4}}{2}-\frac{{\mu}_{1}-{\mu}_{2}}{2}.\phantom{\rule{2em}{0ex}}& \hfill \text{(9.4.12)}\end{array}$$These are the weights of the conjugate spinor representation of $SO\left(8\right)$.

Therefore, we learned that the strong-weak duality of the $SU\left(2\right)$ gauge theory with four ﬂavors,

$$q\leftrightarrow {q}^{\prime}=1\u2215q\leftrightarrow {q}^{\u2033}=1-q$$ | (9.4.13) |

are accompanied by the exchange of the representations of the $SO\left(8\right)$ ﬂavor symmetry,

$$VSC$$ | (9.4.14) |

where $V$, $S$, $C$ are eight dimensional irreducible representations (vector, spinor, conjugate spinor) of $SO\left(8\right)$. These exchanges of three irreducible eight dimensional representations are induced by the outer automorphism, and are known as the triality of the group $SO\left(8\right)$, whose Dynkin diagram is also shown in Fig. 9.11. This triality was originally found in [3]. The exposition in this subsection followed the one given in [8].

The Higgs branch of this system can be studied in any of these descriptions. Originally we have hypermultiplets ${q}_{I}^{a}$, where $a=1,2$ and $I=1,\dots ,8$. The gauge invariant combination is

$${M}_{\left[IJ\right]}={q}_{I}^{a}{q}_{J}^{b}{\mathit{\epsilon}}_{ab}.$$ | (9.4.15) |

In the dual, we have hypermultiplets ${\stackrel{\u0303}{q}}_{\u0128}^{\xe3}$, where $\xe3=1,2$ are for dual $SU\left(2\right)$ and $\u0128=1,\dots ,8$ are for the spinor representation of the $SO\left(8\right)$ ﬂavor symmetry. The basic gauge invariant is then

Both ${M}_{\left[IJ\right]}$ and ${\stackrel{\u0303}{M}}_{\left[\u0128\stackrel{\u0303}{J}\right]}$ are in the adjoint representation of $SO\left(8\right)$, and can be naturally identiﬁed using the outer automorphism of $SO\left(8\right)$. We can check that the constraints satisﬁed by ${M}_{\left[IJ\right]}$ and ${\stackrel{\u0303}{M}}_{\left[\u0128\stackrel{\u0303}{J}\right]}$ are invariant under the outer automorphism. This shows that the Higgs branch are the same as complex spaces.