Let us ﬁrst see concretely how $q$ and ${\tau}_{UV}$ are related. This can be done by computing $a$ and ${a}_{D}$ assuming $\left|q\right|\ll 1$. As always, we put the $A$-cycle at $\left|z\right|=c$, where $\left|q\right|\ll c\ll 1$. Then we easily have

$$a=\frac{1}{2\pi i}{\oint}_{A}x\frac{dz}{z}\sim \sqrt{u}.$$ | (9.2.1) |

The branch points of $\lambda $ are near $1$ and $q$ anyway, and therefore

$${a}_{D}=\frac{1}{2\pi i}{\oint}_{B}x\frac{dz}{z}\sim \frac{1}{2\pi i}2{\int}_{1}^{q}a\frac{dz}{z}=\frac{2a}{2\pi i}logq.$$ | (9.2.2) |

Then

$${\tau}_{U\left(1\right)}=\frac{\partial {a}_{D}}{\partial a}\simeq \frac{2}{2\pi i}logq,$$ | (9.2.3) |

or equivalently

$$q\sim {e}^{2\pi i{\tau}_{UV}}$$ | (9.2.4) |

in the limit of weak coupling; note our convention that ${\tau}_{U\left(1\right)}\sim 2{\tau}_{UV}$. This relation is often written as

$${q}_{C}={e}^{2\pi i{\tau}_{UV,C}}$$ | (9.2.5) |

with an equality. This should be regarded as a nonperturbative deﬁnition of the renormalization and regularization scheme of ${\tau}_{UV}$. Here we added a subscript $C$ to both $q$ and ${\tau}_{UV}$, in order to emphasize that the coupling ${q}_{C}$ is given by the data on the ultraviolet curve $C$.

Another common nonperturbative deﬁnition of the UV coupling constant is to use the low-energy $U\left(1\right)$ coupling ${\tau}_{U\left(1\right)}$ in the limit when the Coulomb vev is very large $\left|u\right|\gg \left|{\stackrel{\u0303}{\mu}}_{i}\right|$:

$${\tau}_{UV,\Sigma}:=\frac{1}{2}\underset{\left|u\right|\to \infty}{lim}{\tau}_{U\left(1\right)}$$ | (9.2.6) |

This should isolate the $SU\left(2\right)$ coupling whose running is stopped at a very large scale given by the Coulomb vev, and can be read oﬀ from the complex structure of the Seiberg-Witten curve $\Sigma $. That is why we used the subscript $\Sigma $ here. Let us also deﬁne

$${q}_{\Sigma}={e}^{2\pi i{\tau}_{UV,\Sigma}}.$$ | (9.2.7) |

This is also a perfectly good scheme, related to the one in (9.2.5) via a ﬁnite renormalization.

To explicitly determine the ﬁnite renormalization, we note that the Seiberg-Witten curve $\Sigma $ in the $\left|u\right|\to \infty $ limit is just the torus which is a double-cover of $C$ branched at $z=0,1,{q}_{C},\infty $. Then ${\tau}_{U\left(1\right)}$ in (9.2.6) is given by the complex structure of this $\Sigma $. From a basic result in the theory of elliptic functions, we ﬁnd

This means that ${\tau}_{UV,C}$ and ${\tau}_{UV,\Sigma}$ are related by a constant shift of its imaginary part plus instanton corrections.

For more extensive discussions on the non-perturbative ﬁnite renormalization, see e.g. Sec. 3.4 and Sec. 3.5 of [18].

Next, let us study mass parameters. Recall that $\lambda $ has mass dimension 1, as its integral give the mass of BPS particles. This means that the ﬁve coeﬃcients of the quartic polynomial $P\left(z\right)$ are of mass dimension two. We can identify these ﬁve coeﬃcients with some combinations of ﬁve parameters ${\mu}_{i=1,2,3,4}$ and $u$. The physical mass parameters are the residues at the poles of $\lambda $. Fixing ${\mu}_{i}$ ﬁxes four linear combinations of the coeﬃcients of $P\left(z\right)$. The sole linear combination which does not change the coeﬃcients of the double poles at $z=0,q,1,\infty $ can be identiﬁed with the parameter $u$. Explicitly, we can write

$${\varphi}_{2}\left(z\right)=\frac{{P}_{0}\left(z\right)}{{\left(z-q\right)}^{2}{\left(z-1\right)}^{2}}\frac{d{z}^{2}}{{z}^{2}}+\frac{u}{\left(z-1\right)\left(z-q\right)}\frac{d{z}^{2}}{z}$$ | (9.2.9) |

where ${P}_{0}\left(z\right)$ is independent of $u$.

Let us now go back to the original curve and study the poles of $\lambda $. We can compute them from (9.1.5) rather easily when the system is weakly coupled, $\left|q\right|\ll 1$. The residues are $\sim {\stackrel{\u0303}{\mu}}_{1,2}$ at $z=0$ and $\sim {\stackrel{\u0303}{\mu}}_{3,4}$ at $z=\infty $. When we went from $\stackrel{\u0303}{x}$ to $x$, we subtracted $\u2663\u22152$ from $x$. We see that the residues are given by

where

$${\mu}_{i}={\stackrel{\u0303}{\mu}}_{i}+O\left(q\right).$$ | (9.2.11) |

The variables ${\stackrel{\u0303}{\mu}}_{i}$ enter rather naturally the Seiberg-Witten curve we guessed in Sec. 6.4.4, whereas the variables ${\mu}_{i}$ enter the BPS mass formula. We see that they are related by a ﬁnite renormalization.

To understand the combinations in (9.2.10) better, it is helpful to consider the $\mathcal{\mathcal{N}}=1$ superpotential. With four doublet hypermultiplets, we have

$$W=\sum _{i}\left({Q}_{i}\Phi {\stackrel{\u0303}{Q}}^{i}+{\mu}_{i}{Q}_{i}{\stackrel{\u0303}{Q}}^{i}\right).$$ | (9.2.12) |

We combine $\left({Q}_{i},{\stackrel{\u0303}{Q}}^{i}\right)$ for $i=1,2,3,4$ to ${q}_{I}$ with $I=1,\dots ,8$. Then the same term becomes

$$W\propto {q}_{I}^{a}{q}_{J}^{b}{\Phi}_{ab}{\delta}^{IJ}+{q}_{I}^{a}{q}_{J}^{b}{\mathit{\epsilon}}_{ab}{\mu}^{IJ}$$ | (9.2.13) |

where ${\mu}^{IJ}$ is a constant matrix with $SO\left(8\right)$ antisymmetric index:

Under the decomposition

$$SO\left(8\right)\supset SO\left(4\right)\times SO\left(4\right)\simeq SU{\left(2\right)}_{A}\times SU{\left(2\right)}_{B}\times SU{\left(2\right)}_{C}\times SU{\left(2\right)}_{D},$$ | (9.2.15) |

the entries of $SO\left(8\right)$ antisymmetric matrix (9.2.14) decomposes to

which are exactly the residues we found in (9.2.10) at $z=0,q,1$ and $=\infty $, respectively. We can regard then that the singularity at $z=0$ carries the $SU{\left(2\right)}_{A}$ symmetry, and that the residue of $\lambda $ there is the mass parameter associated to this $SU{\left(2\right)}_{A}$ symmetry; similarly for $SU{\left(2\right)}_{B}$ at $z=q$, $SU{\left(2\right)}_{C}$ at $z=1$, and $SU{\left(2\right)}_{D}$ at $z=\infty $.

We call these structures the punctures. From the 6d point of view, we consider a puncture at $z=0$ as a four-dimensional object extending along the Minkowski space ${\mathbb{R}}^{3,1}$, which somehow carries an $SU\left(2\right)$ ﬂavor symmetry on it. We will see various other types of punctures below. To distinguish this one from them, we will call this a regular $SU\left(2\right)$ puncture.