We found that writing the Seiberg-Witten curve of the ${N}_{f}=4$ theory in the form

$${\lambda}^{2}-\varphi \left(z\right)=0$$ | (9.6.1) |

helps greatly in understanding the structure of the duality. Let us apply this idea to the curves for theories with less number of ﬂavors, ${N}_{f}<4$.

First, consider the curve of the pure theory,

$$\frac{{\Lambda}^{2}}{z}+{\Lambda}^{2}z={x}^{2}-u,\phantom{\rule{2em}{0ex}}\text{with}\phantom{\rule{1em}{0ex}}\lambda =x\frac{dz}{z}.$$ | (9.6.2) |

In terms of $\lambda $, this can be written as

$${\lambda}^{2}-\varphi \left(z\right)=0,\phantom{\rule{2em}{0ex}}\varphi \left(z\right)=\left(\frac{{\Lambda}^{2}}{z}+u+{\Lambda}^{2}z\right)\frac{d{z}^{2}}{{z}^{2}}.$$ | (9.6.3) |

We see that the quadratic diﬀerential $\varphi \left(z\right)$ has singularities worse than those in the ${N}_{f}=4$ theory: they now have order three poles at $z=0$ and $=\infty $. We can depict the situation of the curve as in the upper row of Fig. 9.17. There, the roman numeral III shows that $\varphi \left(z\right)$ has a third order pole at the puncture. The singularities of $\varphi \left(z\right)$ with higher poles form a new class of punctures, which we call wild $SU\left(2\right)$ punctures.

As an extension of the trivalent diagram encoding the UV Lagrangian, let us introduce the notation that an edge stands for an $\mathcal{\mathcal{N}}=2$ $SU\left(2\right)$ vector multiplet, and the black square at one end means that we do not introduce any hypermultiplet. Then the translation from the diagram representing the UV Lagrangian to the ultraviolet curve can be simply seen, as also shown in Fig. 9.17.

Second, consider the curve of the ${N}_{f}=1$ theory,

This can be written as

where $\lambda =xdz\u2215z$ is shifted from $\stackrel{\u0303}{\lambda}$. We ﬁnd that the singularity at $z=0$ changes to a pole of order 4. The Lagrangian and its Seiberg-Witten solution can be concisely summarized as in the lower row of Fig. 9.17. The edge stands for an $SU\left(2\right)$ gauge group. A black square on one side means that we do not have any hypermultiplet there. A blue blob on another side means that we introduce one hypermultiplet in the doublet. The solution is obtained by associating to a black square by a sphere with a third order pole, denoted by III, and by similarly associating to a blue blob a sphere with a fourth order pole, denoted by IV, and ﬁnally connecting them by a tube. Note that a fourth order pole has its own $SU\left(2\right)$ ﬂavor symmetry and an associated mass parameter.

Summarizing, we consider a sphere with a regular puncture and a wild puncture of pole order III as an empty theory, and a sphere with a regular puncture and a wild puncture of pole order IV as a theory of decoupled doublet hypermultiplet, as shown in Fig. 9.17. Connecting the regular punctures with a tube, we ﬁnd the ultraviolet curves of less ﬂavors.

This generalization allows us to ﬁnd the Seiberg-Witten solutions to a huge class of $\mathcal{\mathcal{N}}=2$ theories whose gauge group is a product of copies of $SU\left(2\right)$. For example, consider a UV Lagrangian theory with gauge group $SU{\left(2\right)}^{4}$ described by the left hand side of Fig. 9.18. In words, we ﬁrst take three copies of bifundamental hypermultiplets,

$${Q}_{aiu},\phantom{\rule{1em}{0ex}}{Q}_{us\alpha}^{\prime},\phantom{\rule{1em}{0ex}}{Q}_{\alpha xm}^{\u2033}.$$ | (9.6.6) |

We showed in the ﬁgure how the indices are assigned to the edges of the trivalent diagram. We emphasized the edges corresponding to the dynamical gauge groups by making them thicker. In words, the indices $a,i,m$ are for $SU{\left(2\right)}_{A,B,C}$ ﬂavor symmetries. An $SU\left(2\right)$ gauge multiplet couples to the index $u$, with exponentiated coupling constant $q$, another $SU\left(2\right)$ gauge multiplet to the index $\alpha $, with exponentiated coupling constant ${q}^{\prime}$. We introduce another $SU{\left(2\right)}_{1}$ gauge multiplet which couples to the index $s$ corresponding to the black square, and ﬁnally another $SU{\left(2\right)}_{2}$ gauge multiplet which couples to the index $x$, with additional ${N}_{f}=1$ hypermultiplet $\left({Q}^{\prime \prime \prime \prime}{\phantom{\rule{0.0pt}{0ex}}}_{x},{\stackrel{\u0303}{Q}}^{\prime \prime \prime \prime}{\phantom{\rule{0.0pt}{0ex}}}^{x}\right)$. We can write down the Lagrangian if required, but now we see how concise the trivalent diagram summarizes its structure.

Its Seiberg-Witten solution can be immediately obtained. It is given by

$${\lambda}^{2}-\varphi \left(z\right)=0,$$ | (9.6.7) |

where $\varphi \left(z\right)$ has three order two poles, one order three pole and ﬁnally an order four pole. Putting them at $z=0,1,{z}_{0}$, ${z}_{1}$ and at $\infty $ respectively, we see that $\varphi \left(z\right)$ has the form

$$\varphi \left(z\right)=\frac{P\left(z\right)}{{z}^{2}{\left(z-1\right)}^{2}{\left(z-{z}_{0}\right)}^{2}{\left(z-{z}_{1}\right)}^{3}}d{z}^{2}$$ | (9.6.8) |

where $P\left(z\right)$ is a polynomial. To have an order four pole at $w=1\u2215z=0$, $P\left(z\right)$ is seen to be a degree-9 polynomial. Among the ten coeﬃcients, three are mass parameters for $SU{\left(2\right)}_{A,B,C}$ ﬂavor symmetry, one is the scale of $SU{\left(2\right)}_{1}$ for the black blob, another is the scale of $SU{\left(2\right)}_{2}$, and another for the mass parameter of the additional ${N}_{f}=1$ ﬂavor for $SU{\left(2\right)}_{2}$. The four remaining linear combinations can be identiﬁed with the four Coulomb branch parameters ${u}_{i}=\u27e8tr\phantom{\rule{0.3em}{0ex}}{\Phi}_{i}^{2}\u22152\u27e9$. This is not a conformal theory: there are two dynamical scales $\Lambda $ and ${\Lambda}^{\prime}$. Still, we immediately see from the structure of the ultraviolet curve that there are S-dualities exchanging the three regular punctures at $A$, $B$ and $C$.