Let us now take the limit $q\to 0$ to decouple the gauge $SU\left(2\right)$, see Fig. 9.6. On the left hand side, we have a sphere parameterized by $z$, with four points at $z=0,q,1$ and $\infty $. On the right hand side, we have two spheres, parameterized by ${z}^{\prime}$ and ${z}^{\u2033}$. We put the points $A$, $B$, $G$ on the ﬁrst sphere, at ${z}^{\prime}=0$, $1$ and $\infty $, and then the points ${G}^{\prime}$, $C$, $D$ on the second sphere, at ${z}^{\u2033}=0$, $1$ and $\infty $. Then we glue the neighborhoods of $G$ and ${G}^{\prime}$ by declaring

$${z}^{\prime}{z}^{\u2033}=q.$$ | (9.3.1) |

Deﬁning $z={z}^{\u2033}$, we see that four points $A,B,C,D$ are exactly as in the ﬁrst description. In this limit, around the tube connecting $G$ and ${G}^{\prime}$, $\lambda \simeq \pm adz\u2215z$. Then in the sphere containing $A$, $B$ and $G$, we have three singularities, with residues of $\lambda $ given by

$$\pm \frac{{\mu}_{1}+{\mu}_{2}}{2},\phantom{\rule{1em}{0ex}}\pm \frac{{\mu}_{1}-{\mu}_{2}}{2},\phantom{\rule{1em}{0ex}}\pm a,$$ | (9.3.2) |

each corresponding to the symmetry $SU{\left(2\right)}_{A}$, $SU{\left(2\right)}_{B}$ and $SU{\left(2\right)}_{G}$, respectively. Here $SU{\left(2\right)}_{G}$ was originally the gauge symmetry.

We were talking about the ${N}_{f}=4$ theory. Then each of the sphere with three punctures should be associated to the ${N}_{f}=2$ hypermultiplet system, see Fig. 9.7; note that this is not coupled to any gauge group. Let us recall the structure of the hypermultiplets again. We start from two hypermultiplets $\left({Q}_{i}^{a},{\stackrel{\u0303}{Q}}_{a}^{i}\right)$ in the doublet of $SU\left(2\right)$, $i=1,2$ and $a=1,2$. We combine them to ${q}_{I}^{a}$, $a=1,2$ and $I=1,\dots ,4$, making $SU\left(2\right)\times SO\left(4\right)$ symmetry manifest. We then decompose the $SO\left(4\right)$ index $I$ into the pair $\left(\alpha ,u\right)$ where $\alpha =1,2$ and $u=1,2$: we have the trifundamental ${q}_{a\alpha u}$. The mass term for this hypermultiplet is

where

Then $\left(a,b\right)$ are the indices for $SU{\left(2\right)}_{G}$, $\left(\alpha ,\beta \right)$ for $SU{\left(2\right)}_{A}$, and $\left(u,v\right)$ for $SU{\left(2\right)}_{B}$. The physical masses of these ﬁelds are given by

$$\pm a\pm \frac{{\mu}_{1}-{\mu}_{2}}{2}\pm \frac{{\mu}_{1}+{\mu}_{2}}{2}=\left\{\pm a\pm {\mu}_{1},\pm a\pm {\mu}_{2}\right\}.$$ | (9.3.5) |

which are the masses for the two doublets of $SU\left(2\right)$ with bare masses ${\mu}_{1,2}$.

The curve of the system, shown in Fig. 9.7 is given by

$${\lambda}^{2}-\varphi \left(z\right)=0,$$ | (9.3.6) |

where $\varphi \left(z\right)$ has the asymptotic behavior

at $z=0$, $z=1$, $z=\infty $ respectively. Here $w=1\u2215z$ as always, and we set $\mu =a$, $\stackrel{\u0303}{\mu}=\left({\mu}_{1}-{\mu}_{2}\right)\u22152$ and $\widehat{\mu}=\left({\mu}_{1}+{\mu}_{2}\right)\u22152$. Note that these asymptotic conditions uniquely ﬁx the quadratic diﬀerential $\varphi \left(z\right)$ to be

$$\varphi \left(z\right)=\frac{{\mu}^{2}{z}^{2}+\left({\widehat{\mu}}^{2}-{\stackrel{\u0303}{\mu}}^{2}-{\mu}^{2}\right)z+{\stackrel{\u0303}{\mu}}^{2}}{{z}^{2}{\left(z-1\right)}^{2}}d{z}^{2}$$ | (9.3.8) |

As was discussed before, the BPS particles of this system can be found by solving the BPS equation (6.1.6)

$$Arg\phantom{\rule{0.3em}{0ex}}\frac{\lambda}{ds}={e}^{i\mathit{\theta}}$$ | (9.3.9) |

for a given $\mathit{\theta}$. As $\varphi \left(z\right)$ given above has two branch points only, the solution to the BPS equation should start from one and end on the other. A computer simulation shows that there are always four and only four such solutions, corresponding to the hypermultiplets with masses given in (9.3.5).