The trifundamental hypermultiplet consisting of $\mathcal{\mathcal{N}}=1$ chiral multiplets ${q}_{a\alpha u}$ for $a,\alpha ,u=1,2$ played the central role in the analysis so far. Let us introduce a shorthand notation for it, by representing it by a trivalent vertex with labels $A$, $B$, $C$ as in Fig. 9.12, signifying the symmetries $SU{\left(2\right)}_{A}$, $SU{\left(2\right)}_{B}$, $SU{\left(2\right)}_{C}$, acting on the indices $a$, $\alpha $, $u$ respectively.

The ultraviolet curve for this system is given by a sphere with three punctures $A$, $B$, $C$, and the Seiberg-Witten curve is given by

$$\Sigma :\phantom{\rule{2em}{0ex}}{\lambda}^{2}-\varphi \left(z\right)=0$$ | (9.5.1) |

where $\varphi \left(z\right)$ is given by the condition that the coeﬃcients of the double poles are given by ${\mu}_{A}^{2}$, ${\mu}_{B}^{2}$, ${\mu}_{C}^{2}$ at each of the punctures $A,B,C$, as in (9.3.7).

Now, the $SU\left(2\right)$ theory with four ﬂavors can be obtained by taking two copies of trifundamentals, and coupling an $SU\left(2\right)$ gauge multiplet to them. We denote it by taking two trivalent vertices, and connecting them by a line, as shown in Fig. 9.12. We put the exponentiated coupling $q$ on the connecting line.

Starting with this trivalent diagram, we can easily write down the Lagrangian of the theory. The free parameters in the theory are the mass parameters ${\mu}_{A,B,C,D}$ and the UV coupling $q$. The ultraviolet curve of this system is given by a sphere with four punctures $A$, $B$, $C$, $D$, and the Seiberg-Witten curve is given by

$$\Sigma :\phantom{\rule{2em}{0ex}}{\lambda}^{2}-\varphi \left(z\right)=0$$ | (9.5.2) |

where $\varphi \left(z\right)$ is given by the condition that its residues are given by ${\mu}_{X}^{2}$ at each of the punctures $X=A,B,C,D$ as in (9.2.10). The triality of the $SU\left(2\right)$ theory with four ﬂavors, shown already in Fig. 9.11, can be depicted in terms of the trivalent diagrams as in Fig. 9.13.

This way, we can regard the trivalent diagram as a shorthand to represent the UV Lagrangian. The Seiberg-Witten solution to this given UV Lagrangian theory is given just by replacing each trivalent vertex with a three punctured sphere, and a connecting line with a connecting tube. This is a surprisingly concise method to obtain the Seiberg-Witten solutions to $\mathcal{\mathcal{N}}=2$ gauge theories.

Let us see a few examples. First, take one trivalent vertex, and connect two out of the three lines by an edge, see Fig. 9.14. We start from a trifundamental described by $\mathcal{\mathcal{N}}=1$ chiral multiplets ${q}_{a\alpha u}$, but we couple the same $SU\left(2\right)$ gauge multiplet to the index $a$ and $\alpha $. Then the combination $\left(a,\alpha \right)$ is in the tensor product of two spin-$1\u22152$ representations. Therefore we can split it into a triplet and a singlet, with additional index $u=1,2$:

$${q}_{a\alpha u}\to {q}_{iu}^{\prime},{q}_{u}^{\u2033},$$ | (9.5.3) |

where $i=1,2,3$ is the index for the triplet of $SU\left(2\right)$.

In total, we just have one full hypermultiplet in the triplet, and another full hypermultiplet in the singlet which is completely decoupled. Therefore this is essentially the $\mathcal{\mathcal{N}}={2}^{\ast}$ $SU\left(2\right)$ theory, or equivalently the $\mathcal{\mathcal{N}}=4$ $SU\left(2\right)$ theory with mass deformation to the hypermultiplet in the adjoint representation. The adjoint mass $\mu $ is associated to the remaining one $SU\left(2\right)$ ﬂavor symmetry.

Its Seiberg-Witten solution is given by connecting two punctures of a three-punctured sphere by a tube. As shown in Fig. 9.14, the ultraviolet curve is a torus with one puncture. The Seiberg-Witten curve is then

$${\lambda}^{2}-\varphi \left(z\right)=0$$ | (9.5.4) |

where $z$ is now a coordinate of the torus, which we take to be the complex plane with the identiﬁcation $z\sim z+1\sim z+\tau $. As the origin of the coordinate is arbitrary, we put the puncture at the origin. The $\varphi \left(z\right)$ is given by the condition that it has a double pole with a given strength at $z=0$. This uniquely ﬁxes the form of $\varphi \left(z\right)$ to be

$$\varphi \left(z\right)=\left({\mu}^{2}\wp \left(z;\tau \right)+u\right)d{z}^{2}$$ | (9.5.5) |

where $\wp $ is the Weierstraß function, and $u$ is the Coulomb branch vev $u=\u27e8tr\phantom{\rule{0.3em}{0ex}}{\Phi}^{2}\u27e9\u22152$.

Now it is clear that the theory at the coupling given by $\tau $ and the same theory at the coupling given by ${\tau}^{\prime}=-1\u2215\tau $ are equivalent after exchanging the monopoles and the adjoint quarks. The space of the coupling can be identiﬁed with the moduli space ${\mathcal{\mathcal{M}}}_{1}$ of the tori, i.e genus-1 Riemann surfaces, which is given by

$${\mathcal{\mathcal{M}}}_{1}=\mathbb{H}\u2215SL\left(2,\mathbb{Z}\right)$$ | (9.5.6) |

where $\mathbb{H}$ is the upper half plane where $\tau $ takes the value in, and $SL\left(2,\mathbb{Z}\right)$ is the modular group exchanging the edges of the torus. The duality group can be identiﬁed with the modular group.

As another example, take three trivalent vertices and connect them as shown in Fig. 9.15. The leftmost trivalent vertex counts as ${N}_{f}=2$ ﬂavors for $SU{\left(2\right)}_{1}$, and the rightmost one counts as ${N}_{f}=2$ ﬂavors for $SU{\left(2\right)}_{2}$. In addition, we have a hypermultiplet coming from the central trivalent vertex, ${q}_{aiu}$ where $a$ is for $SU{\left(2\right)}_{1}$ and $i$ is for $SU{\left(2\right)}_{2}$. The remaining index $u=1,2$ is an index for the ﬂavor symmetry. In older literature it was more customary to denote this hypermultiplet charged under $SU{\left(2\right)}_{1}\times SU{\left(2\right)}_{2}$ using $\mathcal{\mathcal{N}}=1$ chiral multiplets $\left({Q}_{a}^{i},{\stackrel{\u0303}{Q}}_{i}^{a}\right)$ which are

$${Q}_{a}^{i}={q}_{aju=1}{\mathit{\epsilon}}^{ij},\phantom{\rule{2em}{0ex}}{\stackrel{\u0303}{Q}}_{i}^{a}={q}_{biu=2}{\mathit{\epsilon}}^{ab}.$$ | (9.5.7) |

This is usually called the bifundamental multiplet charged under $SU{\left(2\right)}_{1}\times SU{\left(2\right)}_{2}$.

The Seiberg-Witten solution to this theory is easily found, as shown in Fig. 9.15. We start from three spheres, described by complex coordinates ${z}_{1}$, ${z}_{2}$, and ${z}_{3}$. The punctures $A,B$ are at ${z}_{1}=0,1$; the puncture $C$ is at ${z}_{2}=1$; the punctures $D,E$ are at ${z}_{3}=1,\infty $, respectively. To connect ${z}_{1}=\infty $ and ${z}_{2}=0$, we introduce ${w}_{1}=1\u2215{z}_{1}$ and require the relation ${w}_{1}{z}_{2}={q}^{\prime}$. This simply means that we have via ${z}_{1}={q}^{\prime}{z}_{2}$. Similarly, by connecting ${z}_{2}=\infty $ and ${z}_{3}=0$, we have ${z}_{2}=q{z}_{3}$. Now we introduce $z={z}_{3}$ to describe the coordinate on the resulting sphere with ﬁve punctures. Then the punctures are at $z=0$, $q{q}^{\prime}$, $q$, $1$ and $\infty $, each representing an $SU\left(2\right)$ ﬂavor symmetry which we call $SU{\left(2\right)}_{A,B,C,D,E}$ respectively. The gauge couplings of $SU{\left(2\right)}_{1}\times SU{\left(2\right)}_{2}$ can be identiﬁed with $q$ and ${q}^{\prime}$.

Let us denote the mass parameters associated to the ﬂavor symmetries by ${\mu}_{A,B,C,D,E}$. The Seiberg-Witten curve is

$${\lambda}^{2}-\varphi \left(z\right)=0$$ | (9.5.8) |

where $\varphi \left(z\right)$ needs to satisfy the asymptotic behavior

$$\varphi \left(z\right)\sim \frac{{\mu}_{X}^{2}}{{z}_{X}^{2}}d{z}_{X}^{2}$$ | (9.5.9) |

where ${z}_{X}$ for $X=A,B,C,D,E$ is a local coordinate on the ultraviolet curve such that the puncture $X$ is at ${z}_{X}=0$. From the conditions at $A$, $B$, $C$, $D$, we ﬁnd that $\varphi $ is given by

$$\varphi \left(z\right)=\frac{P\left(z\right)}{{z}^{2}{\left(z-1\right)}^{2}{\left(z-q\right)}^{2}{\left(z-q{q}^{\prime}\right)}^{2}}d{z}^{2}$$ | (9.5.10) |

where $P\left(z\right)$ is a polynomial. To impose the condition at $E$, we go to the coordinate $w=1\u2215z$. For $\varphi \left(z\right)$ to behave as $\sim d{w}^{2}\u2215{w}^{2}$, $P\left(z\right)$ can have terms of up to ${z}^{6}$. We see that $\varphi \left(z\right)$ has seven coeﬃcients. Five combinations are mass parameters, and two linear combinations that do not shift the coeﬃcients of the double poles are the Coulomb branch parameters $u=\u27e8tr\phantom{\rule{0.3em}{0ex}}{\Phi}^{2}\u27e9\u22152$ and ${u}^{\prime}=\u27e8tr\phantom{\rule{0.3em}{0ex}}{\Phi}^{\prime}{\phantom{\rule{0.0pt}{0ex}}}^{2}\u27e9\u22152$ of two gauge multiplets $SU{\left(2\right)}_{1,2}$. From the Seiberg-Witten solution, we see that this theory has strong-weak coupling dualities where ﬁve ﬂavor symmetry groups $SU{\left(2\right)}_{A,B,C,D,E}$ can be arbitrarily permuted, with an appropriate change of the couplings $\left(q,{q}^{\prime}\right)$ of the two gauge groups. This extended duality was ﬁrst found in [56].

As the third example, let us take two trivalent vertices and connect them with three edges. There are two topologically distinct ways to do this, as shown on the left hand side of Fig. 9.16.

The upper theory is an $SU{\left(2\right)}_{l}\times SU{\left(2\right)}_{m}\times SU{\left(2\right)}_{r}$ gauge theory. There are half-hypermultiplets which are in

$$3\otimes 2\otimes 1,\phantom{\rule{2em}{0ex}}1\otimes 2\otimes 3$$ | (9.5.11) |

and one full hypermultiplet charged under $SU{\left(2\right)}_{m}$. Note that the trivalent-graph construction does not allow us to consider theories with non-zero mass term for this last full hypermultiplet. The lower theory is an $SU{\left(2\right)}_{1}\times SU{\left(2\right)}_{2}\times SU{\left(2\right)}_{3}$ theory with two half-hypermultiplets in the trifundamental representation. Again, the trivalent-graph construction does not allow us to introduce non-zero mass term for this full hypermultiplet in the trifundamental.

The Seiberg-Witten solution is again easily obtained. To obtain the ultraviolet curve, we replace two trivalent vertices with three-punctured spheres, and connect pairs of punctures with tubes. We see that both are given by a smooth genus-2 surface. The Seiberg-Witten curve is a further double cover given by

$${\lambda}^{2}-\varphi \left(z\right)=0$$ | (9.5.12) |

where $z$ is a complex coordinate of the genus-2 surface, and $\varphi \left(z\right)$ is a smooth quadratic diﬀerential on it. The space of quadratic diﬀerentials on a ﬁxed genus-2 surface is complex three dimensional, which we identify with the Coulomb branch vevs ${u}_{i}=\u27e8tr\phantom{\rule{0.3em}{0ex}}{\Phi}_{i}^{2}\u27e9\u22152$.

Now it is clear that we can continuously deform the upper theory to the lower theory by tuning the gauge couplings. The non-perturbative space of couplings can be identiﬁed with the moduli space ${\mathcal{\mathcal{M}}}_{2}$ of genus-2 Riemann surfaces, which is complex three dimensional. The duality group is identiﬁed with the mapping class group ${\mathcal{\mathcal{G}}}_{2}$ of the genus-2 surface, and ${\mathcal{\mathcal{M}}}_{2}={\mathcal{\mathcal{T}}}_{2}\u2215{\mathcal{\mathcal{G}}}_{2}$ where ${\mathcal{\mathcal{T}}}_{2}$ is the Teichmüller space of the genus-2 Riemann surface, compare the genus-1 case (9.5.6).

Now a somewhat surprising mathematical fact is that
${\mathcal{\mathcal{T}}}_{2}$
is equivalent to three copies of the upper half plane
${\mathbb{H}}^{3}$ in the smooth sense, but not
in the holomorphic sense^{12} :

Naive perturbative analysis tells us that the space of the couplings of $SU{\left(2\right)}^{3}$ is just three copies of the upper half plane:

$$\left({\tau}_{1},{\tau}_{2},{\tau}_{3}\right)\in {\mathbb{H}}^{3}$$ | (9.5.14) |

including the complex structure. Therefore, we ﬁnd that the non-perturbative corrections can make a rather drastic change in the complex structure of the parameter space of supersymmetric theories.

We learned that writing the Seiberg-Witten curve in the form

$$\Sigma :\phantom{\rule{2em}{0ex}}{\lambda}^{2}-\varphi \left(z\right)=0$$ | (9.5.15) |

is very useful for the understanding of the system. This way of presenting the curve is closely related to the so-called Hitchin system on the ultraviolet curve $C$. Explaining this technique would make another lecture note. A starting point for the reader is e.g. Sec. 3 of [51].

Let us at least present a very crude aspect of it. We consider a meromoprhic one-form $\phi \left(z\right)$ which is a traceless $2\times 2$ matrix. Then, we can form an equation of the form

$$det\left(\lambda -\phi \left(z\right)\right)=0$$ | (9.5.16) |

for a one-form $\lambda $. Identifying (9.5.15) and (9.5.16), we ﬁnd

$$\frac{1}{2}tr\phantom{\rule{0.3em}{0ex}}\phi {\left(z\right)}^{2}=\varphi \left(z\right).$$ | (9.5.17) |

This matrix ﬁeld $\phi \left(z\right)$ is
called the Hitchin ﬁeld.^{13}
It is possible to understand the existence of such a complex adjoint ﬁeld on the ultraviolet curve
using string duality, but explaining it is outside the aim of this lecture note.

The condition on the ﬁeld $\varphi \left(z\right)$ at a puncture at ${z}_{X}=0$ associated to a mass term ${\mu}_{X}$ was given in (9.5.9):

$$\varphi \left(z\right)={\mu}_{X}^{2}\frac{d{z}_{X}^{2}}{{z}_{X}^{2}}+\text{(lesssingularterms)}.$$ | (9.5.18) |

This translates to the condition for the Hitchin ﬁeld $\phi \left(z\right)$ given by

Here, the symbol $\sim $ means that the terms on the left and the right are conjugate.

Now, let us consider what happens when we turn oﬀ ${\mu}_{X}$ to zero. In the description (9.5.18), we ﬁnd that the boundary condition becomes

$$\varphi \left(z\right)=c\frac{d{z}_{X}^{2}}{{z}_{X}}+\text{(lesssingularterms)}$$ | (9.5.20) |

for some constant $c$. In the description (9.5.19), it is not that the residue just becomes a zero matrix. We note that

as long as ${\mu}_{X}\ne 0$, and we can take the limit ${\mu}_{X}\to 0$ on the right hand side. This means that the massless limit results in the boundary condition of the form

$$\phi \left(z\right)\sim \left(\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)\frac{d{z}_{X}}{{z}_{X}}+\text{(lesssingularterms)}.$$ | (9.5.22) |

Using (9.5.17), one ﬁnds that this reproduces the condition (9.5.20).

At this stage, one might not ﬁnd the advantange of using $\phi \left(z\right)$ instead of $\varphi \left(z\right)$ very much. It turns out, however, that when we discuss a generalization to $SU\left(N\right)$ gauge theories or gauge theories with more complicated gauge groups, it turns out to be crucial.