9.2 Identification of parameters

9.2.1 Coupling constant

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

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

Then

or equivalently

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

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 $\left|u\right|\gg \left|{\tilde{\mu }}_i\right|$:

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

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].

9.2.2 Mass parameters

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

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 $\sim {\tilde{\mu }}_{1,2}$ at $z=0$ and $\sim {\tilde{\mu }}_{3,4}$ at $z=\infty$. When we went from $\tilde{x}$ to $x$, we subtracted $♣∕2$ from $x$. We see that the residues are given by

where

The variables ${\tilde{\mu }}_i$ 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 $\mathcal{𝒩}=1$ superpotential. With four doublet hypermultiplets, we have

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

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

Under the decomposition

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 ${ℝ}^{3,1}$, 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.