In terms of $\mathcal{\mathcal{N}}=1$ chiral multiplets, the hypermultiplet consists of two $SU\left(2\right)$ doublets ${Q}^{a}$ and ${\stackrel{\u0303}{Q}}_{a}$ where $a=1,2$ is the $SU\left(2\right)$ index. There is an $\mathcal{\mathcal{N}}=1$ superpotential

$$W=Q\Phi \stackrel{\u0303}{Q}+\mu Q\stackrel{\u0303}{Q}$$ | (5.1.1) |

where $\mu $ is the bare mass of the hypermultiplet. Classically, $\Phi =diag\left(a,-a\right)$ together with $Q=\stackrel{\u0303}{Q}=0$ still gives supersymmetric vacua. With nonzero $a$, the gauge group is broken to $U\left(1\right)$, and the physical mass of the ﬁelds $Q$ and $\stackrel{\u0303}{Q}$ can be found by explicitly expanding the superpotential above:

We see that the masses are

$$|\pm a\pm \mu |$$ | (5.1.3) |

where we allow all four choices of signs.

In general, the BPS mass formula is

$$\text{mass}\ge |na+m{a}_{D}+f\mu |$$ | (5.1.4) |

where $f$ is the charge under the $U\left(1\right)$ ﬂavor symmetry, under which $Q$ has charge $1$ and $\stackrel{\u0303}{Q}$ has charge $-1$.

From the one-loop running of the coupling constant, we ﬁnd

$$\begin{array}{lll}\hfill \tau \left(a\right)& =2{\tau}_{UV}-\frac{6}{2\pi i}log\frac{a}{{\Lambda}_{UV}}+\cdots \phantom{\rule{2em}{0ex}}& \hfill \text{(5.1.5)}\\ \hfill & =-\frac{6}{2\pi i}log\frac{a}{{\Lambda}_{1}}+\cdots \phantom{\rule{2em}{0ex}}& \hfill \text{(5.1.6)}\end{array}$$in the ultraviolet region. Here we deﬁned

$${\Lambda}_{1}^{6}={\Lambda}_{UV}^{6}{e}^{4\pi i{\tau}_{UV}}$$ | (5.1.7) |

where the subscript $1$ is a reminder that we are dealing with the ${N}_{f}=1$ theory. From this, we can determine the monodromy ${M}_{\infty}$ at inﬁnity acting on $\left(a,{a}_{D}\right)$:

$${M}_{\infty}=\left(\begin{array}{cc}\hfill -1\hfill & \hfill 3\hfill \\ \hfill 0\hfill & \hfill -1\hfill \end{array}\right)$$ | (5.1.8) |

exactly as in the pure $SU\left(2\right)$ case.

To study the strong coupling region of the system, let us ﬁrst consider two extreme cases. When $\left|\mu \right|$ is very big, we expect the running of the coupling to be given roughly as in Fig. 5.1. Namely, at around the scale $\left|\mu \right|$, the ﬁelds $Q$ and $\stackrel{\u0303}{Q}$ decouple, and the system eﬀectively becomes the pure $SU\left(2\right)$ gauge theory, which we studied in the last section. Correspondingly, the structure of the $u$-plane in the region $\left|u\right|\ll \left|{\mu}^{2}\right|$ should be eﬀectively the same with that of the pure $SU\left(2\right)$ theory, with two singularities at $u=\pm 2{\Lambda}_{0}^{2}$.

A rough relation between ${\Lambda}_{0}$ and ${\Lambda}_{1}$ can be read oﬀ from the schematic graph of the running coupling shown in Fig. 5.1. The rightmost segment in the graph is given by

$$\tau \left(E\right)=-\frac{6}{2\pi i}log\frac{E}{{\Lambda}_{1}}$$ | (5.1.9) |

and the middle segment in the graph, representing the eﬀectively pure $SU\left(2\right)$ theory, is

$$\tau \left(E\right)=-\frac{8}{2\pi i}log\frac{E}{{\Lambda}_{0}}.$$ | (5.1.10) |

Equating these two values at $E=\mu $, we obtain

$${\Lambda}_{0}^{4}=\mu {\Lambda}_{1}^{3}.$$ | (5.1.11) |

In addition, we know from (5.1.3) that the quanta of one component of $Q$ and $\stackrel{\u0303}{Q}$ become very light when $\pm a\sim \mu $. This should produce a singularity in the $u$-plane at around $u\simeq {\mu}^{2}$. We therefore expect that the $u$-plane to have three singularities, as shown in Fig. 5.2. Note that local physics at the three singularities, at $u\simeq {\mu}^{2}$ and at $u\simeq \pm 2{\Lambda}_{0}^{2}$, is always just $U\left(1\right)$ gauge theory with one charged hypermultiplet.

In the other extreme when $\mu =0$, we can make use of the discrete R symmetry. The standard R-charge assignment is as follows:

The rotation

$$\lambda \to {e}^{i\phi}\lambda ,\phantom{\rule{2em}{0ex}}{\psi}_{Q,\stackrel{\u0303}{Q}}\to {e}^{-i\phi}{\psi}_{Q,\stackrel{\u0303}{Q}}$$ | (5.1.13) |

is anomalous, but can be compensated by

$${\mathit{\theta}}_{UV}\to {\mathit{\theta}}_{UV}+6\phi .$$ | (5.1.14) |

Therefore $\phi =2\pi \u22156$ is a genuine symmetry, which does

$$\mathit{\theta}\to \mathit{\theta}+2\pi ,\phantom{\rule{1em}{0ex}}\Phi \to {e}^{2\pi i\u22153}\Phi ,\phantom{\rule{1em}{0ex}}u\to {e}^{4\pi i\u22153}u.$$ | (5.1.15) |

This guarantees that singularities in the $u$-plane should appear in triples, related by $12{0}^{\circ}$ rotation. A minimal assumption is then to have exactly three singularities, as shown in Fig. 5.3. Having three singularities is consistent with our previous analysis when $\left|\mu \right|$ was very big. We expect that the situation in Fig. 5.2 will smoothly change into the one in Fig. 5.3 when $\mu $ is adiabatically changed.

Let us denote the monodromies around each of the three singularities by ${M}_{1,2,3}$, see Fig. 5.4. Clearly, we should have

$${M}_{\infty}={M}_{3}{M}_{2}{M}_{1}$$ | (5.1.16) |

where ${M}_{\infty}$ was given in (5.1.8). As the three singularities are related by discrete R-symmetry, they should be conjugate. For example, as shown in Fig. 5.5, we expect ${M}_{2}=Y{M}_{1}{Y}^{-1}$.

A solution is given by

$${M}_{2}={T}^{-1}{M}_{1}{T}^{1},\phantom{\rule{1em}{0ex}}{M}_{3}={T}^{-2}{M}_{1}{T}^{2},$$ | (5.1.17) |

together with

$${M}_{1}=ST{S}^{-1}=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill -1\hfill & \hfill 1\hfill \end{array}\right).$$ | (5.1.18) |

As ${M}_{1}$ found here is the same as ${M}_{+}$ found in the pure case (4.2.9), the local physics close to the singularity is also the same, i.e. it is described by an $\mathcal{\mathcal{N}}=2$ $U\left(1\right)$ gauge theory coupled to one charged hypermultiplet. The same can be said for ${M}_{2}$ and ${M}_{3}$.

For the pure case, we saw that the light charged hypermultiplet in this low energy $U\left(1\right)$ description was a monopole in the original description. Is the same true in this case? It is easier to give a deﬁnitive answer when $\left|\mu \right|$ is very big. Then, the two singularities in the strong coupled region have the same physics as that of the pure $SU\left(2\right)$ theory, and thus we should have light monopoles and dyons there. At the third singularity $u\simeq {\mu}^{2}$, one component of the doublet hypermultiplet $\left(Q,\stackrel{\u0303}{Q}\right)$ becomes very light. For all three singularities, the low-energy description is that of a $U\left(1\right)$ gauge theory coupled to one charged hypermultiplet.

By gradually decreasing $\mu $ to be zero, these three singularities become the three singularities related by the discrete R symmetry. At this stage, it is not possible to say which of the three was originally the one whose light particle came from the doublet hypermultiplet and which two of the three were the ones with monopoles and dyons. This loss of the distinction between the hypermultiplets which were elementary ﬁelds and the hypermultiplets which came from solitons such as monopoles or dyons is somewhat surprising to an eye trained in the classical ﬁeld theory. We will see this more explicitly below, in Fig. 5.10.