Let us come back to the study of the Higgs branch. The equations deﬁning it were given in Sec. 2.2 for the case of $SU\left(N\right)$ gauge theory with ${N}_{f}$ ﬂavors, see (2.2.3) and (2.2.6). Let us write them down for the general case.

Consider an $\mathcal{\mathcal{N}}=2$ gauge theory with gauge group $G$ and a hypermultiplet $\left({Q}^{i},{\stackrel{\u0303}{Q}}_{i}\right)$ in the representation $R$. Here the index $i=1,\dots ,dimR$ is for the hypermultiplet and we use the index $a=1,\dots ,dimG$ for the adjoint representation. The Higgs branch is given by

where ${T}^{a}{\phantom{\rule{0.0pt}{0ex}}}_{i}^{j}$ is the matrix of the algebra of $G$ in the representation $R$.

There is no massless vector multiplet remaining in the generic point of the Higgs branch. From the general analysis in the preceding sections, we know that they form a hyperkäher manifold. The construction (7.3.1) is known as the hyperkähler quotient construction in the literature both in mathematics and in physics, and found originally in [54]. The real dimension of any hyperkähler manifold is always a multiple of four. Let us check this in this situation. Suppose the original hypermultiplets consist of $4m$ real scalars. The D-term condition imposes $dimG$ real constraints, for each $I$, $J$ and $K$. Then we make the identiﬁcation by the action of $G$. Therefore we have

$$4m-3dimG-dimG=4\left(m-dimG\right)$$ | (7.3.2) |

real dimensions after the quotient.

If we are only interested in the holomorphic structure, we can drop the $D$-term equation and instead perform the identiﬁcation by the complexiﬁed gauge group

$$\left\{{Q}^{i}{\stackrel{\u0303}{Q}}_{j}{T}^{a}{\phantom{\rule{0.0pt}{0ex}}}_{i}^{j}=0\right\}/\left(\text{identi\ufb01cationbythecomplexi\ufb01edgaugegroup}\right).$$ | (7.3.3) |

Note that this is a more natural form in the $\mathcal{\mathcal{N}}=1$ superﬁeld formulation, if we do not put the vector superﬁeld into the Wess-Zumino gauge. The basic idea to show the equality of (7.3.3) with (7.3.1) is to minimize $|D{|}^{2}$ within each of the orbit of the complexﬁed gauge group. The minimization condition then gives $D=0$, recovering (7.3.1). This rough analysis also shows that, more precisely speaking, we need to remove the so-called unstable orbits in (7.3.3), in which there is no point where $|D{|}^{2}$ is minimized.

In this approach, we start from $2m$ complex scalars. We then imposes $dimG$ complex constraints and then perform the identiﬁcation by the action of ${G}_{\u2102}$, the complexiﬁed gauge group, removing $dimG$ complex dimensions. We end up with

$$2m-dimG-dimG=2\left(m-dimG\right)$$ | (7.3.4) |

complex scalars in the quotient. This is compatible with what we just found in (7.3.2). If we count the quaternionic dimension, we just have the formula

$$m-dimG.$$ | (7.3.5) |

Let us consider two examples. First, take an $\mathcal{\mathcal{N}}=2$ $U\left(1\right)$ gauge theory with two hypermultiplets $\left({Q}_{i},{\stackrel{\u0303}{Q}}^{i}\right)$ with charge $\pm 1$. Here $i=1,2$. We have $m=2$ and $dimG=1$ in the expressions above, so we expect a complex two-dimensional Higgs branch. First, let us determine the Higgs branch explicitly. The F-term equation is

$${Q}_{1}{\stackrel{\u0303}{Q}}^{1}+{Q}_{2}{\stackrel{\u0303}{Q}}^{2}=0.$$ | (7.3.6) |

Then we have

for some complex numbers $z$, $\stackrel{\u0303}{z}$ and $t$. Then the D-term equation $|{Q}_{1}{|}^{2}+|{Q}_{2}{|}^{2}-|{\stackrel{\u0303}{Q}}^{1}{|}^{2}-|{\stackrel{\u0303}{Q}}_{2}|=0$ says

$$|z{|}^{2}+|\stackrel{\u0303}{z}{|}^{2}=|t{|}^{2}\left(|z{|}^{2}+|\stackrel{\u0303}{z}{|}^{2}\right).$$ | (7.3.8) |

Therefore we see $\left|t\right|=1$. We can use the $U\left(1\right)$ gauge rotation to eliminate $t$ almost completely, by demanding

$$Arg\phantom{\rule{0.3em}{0ex}}z=Arg\phantom{\rule{0.3em}{0ex}}\left(\stackrel{\u0303}{z}t\right).$$ | (7.3.9) |

This still does not ﬁx the $U\left(1\right)$ gauge transformation given by the multiplication by $-1$ on ${Q}_{i}$, ${\stackrel{\u0303}{Q}}^{i}$, sending the pair $\left(z,\stackrel{\u0303}{z}\right)$ to $\left(-z,-\stackrel{\u0303}{z}\right)$. We conclude that the Higgs branch is given by

A not-quite-accurate schematic description is given in Fig. 7.1.

Let us use the complex description (7.3.3) to obtain the same Higgs branch in a diﬀerent way. Instead of identifying points connected by the complexiﬁed gauge group, we can just consider combinations of coordinates which are invariant under it. In this case, ${Q}_{i}$ has charge $+1$ and ${\stackrel{\u0303}{Q}}^{i}$ has charge $-1$. Then, the gauge invariants are ${Q}_{i}{\stackrel{\u0303}{Q}}^{j}$, for arbitrary choices of $i$ and $j$. We need to impose

$${Q}_{1}{\stackrel{\u0303}{Q}}^{1}+{Q}_{2}{\stackrel{\u0303}{Q}}^{2}=0,$$ | (7.3.11) |

too. In total, we ﬁnd three combinations

They satisfy one obvious relation

$$AB={C}^{2}.$$ | (7.3.13) |

With three variables $A$, $B$, $C$ and one relation above, we have complex two-dimensional space. This is the Higgs branch.

This description can also be found starting from the deﬁnition of ${\u2102}^{2}\u2215{\mathbb{Z}}_{2}$ in (7.3.10). Combinations of $z$, $\stackrel{\u0303}{z}$ invariant under $\left(z,\stackrel{\u0303}{z}\right)\leftrightarrow \left(-z,-\stackrel{\u0303}{z}\right)$ are

$$A={z}^{2},\phantom{\rule{1em}{0ex}}B={\stackrel{\u0303}{z}}^{2},\phantom{\rule{1em}{0ex}}C=z\stackrel{\u0303}{z}$$ | (7.3.14) |

which satisfy the same relation (7.3.13). Therefore they are the same spaces as complex manifolds.

As the second example, consider $\mathcal{\mathcal{N}}=2$ $SU\left(2\right)$ gauge theory with ${N}_{f}$ full hypermultiplets in the doublet representations. In terms of $\mathcal{\mathcal{N}}=1$ chiral multiplets, we have

$${Q}_{i}^{a},\phantom{\rule{1em}{0ex}}{\stackrel{\u0303}{Q}}_{a}^{i}\phantom{\rule{2em}{0ex}}\left(a=1,2;\phantom{\rule{1em}{0ex}}i=1,\dots ,{N}_{f}\right)$$ | (7.3.15) |

As the doublet and the anti-doublet representations are the same for $SU\left(2\right)$ gauge theory, we can denote them also as

$${Q}_{I}^{a},\phantom{\rule{2em}{0ex}}\left(a=1,2;\phantom{\rule{1em}{0ex}}I=1,\dots ,2{N}_{f}\right)$$ | (7.3.16) |

which makes $SO\left(2{N}_{f}\right)$ ﬂavor symmetry more manifest.

We have $4{N}_{f}$ complex scalars and $dimSU\left(2\right)=3$. Then the complex dimension of the Higgs branch is

$$4{N}_{f}-2\cdot 3.$$ | (7.3.17) |

So we do not have the Higgs branch for ${N}_{f}=1$, and expect a Higgs branch with complex dimensions $2$, $6$, $10$ for ${N}_{f}=2,3,4$, respectively. Let us study the case ${N}_{f}=2$ in more detail.

Gauge-invariant combinations of ${Q}_{I}^{a}$ are

$${M}_{IJ}={Q}_{I}^{a}{Q}_{J}^{b}{\mathit{\epsilon}}_{ab}.$$ | (7.3.18) |

The left hand side is automatically anti-symmetric under the exchange of $I$ and $J$. The F-term equation is

$${Q}_{I}^{(a}{Q}_{J}^{b}{\delta}^{IJ}=0.$$ | (7.3.19) |

For $SO\left(2{N}_{f}\right)=SO\left(4\right)$, we can split an antisymmetric matrix ${M}_{IJ}$ of $SO\left(4\right)$ into the self-dual and the anti-self-dual parts. Equivalently, using $SO\left(4\right)\simeq SU{\left(2\right)}_{u}\times SU{\left(2\right)}_{v}$, ${M}_{IJ}$ splits into the triplet ${M}_{\left(\alpha \beta \right)}$ of $SU{\left(2\right)}_{u}$ and the triplet ${M}_{\left(\stackrel{\u0307}{\alpha}\stackrel{\u0307}{\beta}\right)}$ of $SU{\left(2\right)}_{v}$, where $\alpha ,\beta =1,2$ and $\stackrel{\u0307}{\alpha},\stackrel{\u0307}{\beta}=1,2$ are doublet indices of $SU{\left(2\right)}_{u,v}$ respectively. The index $I$ itself can be thought of a pair of indices: $I=\left(\alpha \stackrel{\u0307}{\alpha}\right)$. Then the hypermultiplets we are dealing with can be written as

$${Q}_{a\alpha \stackrel{\u0307}{\alpha}},\phantom{\rule{2em}{0ex}}a,\alpha ,\stackrel{\u0307}{\alpha}=1,2$$ | (7.3.20) |

which makes the existence of $SU{\left(2\right)}^{3}$ symmetry manifest. Then

$$\begin{array}{lll}\hfill {M}_{\alpha \beta}& ={Q}_{a\alpha \stackrel{\u0307}{\alpha}}{Q}_{b\beta \stackrel{\u0307}{\beta}}{\mathit{\epsilon}}^{ab}{\mathit{\epsilon}}^{\stackrel{\u0307}{\alpha}\stackrel{\u0307}{\beta}},\phantom{\rule{2em}{0ex}}& \hfill \text{(7.3.21)}\\ \hfill {M}_{\stackrel{\u0307}{\alpha}\stackrel{\u0307}{\beta}}& ={Q}_{a\alpha \stackrel{\u0307}{\alpha}}{Q}_{b\beta \stackrel{\u0307}{\beta}}{\mathit{\epsilon}}^{ab}{\mathit{\epsilon}}^{\alpha \beta}\phantom{\rule{2em}{0ex}}& \hfill \text{(7.3.22)}\end{array}$$are the self-dual and the anti-self-dual parts of ${M}_{IJ}$, respectively. The F-term equation can be written as

$${Q}_{a\alpha \stackrel{\u0307}{\alpha}}{Q}_{b\beta \stackrel{\u0307}{\beta}}{\mathit{\epsilon}}^{\alpha \beta}{\mathit{\epsilon}}^{\stackrel{\u0307}{\alpha}\stackrel{\u0307}{\beta}}=0.$$ | (7.3.23) |

Using this description, it is not very hard to check that

$$\begin{array}{lll}\hfill {M}_{\alpha \beta}{M}_{\gamma \delta}{\mathit{\epsilon}}^{\alpha \gamma}{\mathit{\epsilon}}^{\beta \delta}& =0,\phantom{\rule{2em}{0ex}}& \hfill \text{(7.3.24)}\\ \hfill {M}_{\stackrel{\u0307}{\alpha}\stackrel{\u0307}{\beta}}{M}_{\stackrel{\u0307}{\gamma}\stackrel{\u0307}{\delta}}{\mathit{\epsilon}}^{\stackrel{\u0307}{\alpha}\stackrel{\u0307}{\gamma}}{\mathit{\epsilon}}^{\stackrel{\u0307}{\beta}\stackrel{\u0307}{\delta}}& =0,\phantom{\rule{2em}{0ex}}& \hfill \text{(7.3.25)}\\ \hfill {M}_{\alpha \beta}{M}_{\stackrel{\u0307}{\alpha}\stackrel{\u0307}{\beta}}& =0.\phantom{\rule{2em}{0ex}}& \hfill \text{(7.3.26)}\end{array}$$The structure becomes clearer by deﬁning

$$\begin{array}{lllllll}\hfill A& ={M}_{11},\phantom{\rule{2em}{0ex}}& \hfill B& ={M}_{22},\phantom{\rule{2em}{0ex}}& \hfill C& ={M}_{12}={M}_{21};\phantom{\rule{2em}{0ex}}& \hfill \text{(7.3.27)}\\ \hfill X& ={M}_{\stackrel{\u0307}{1}\stackrel{\u0307}{1}},\phantom{\rule{2em}{0ex}}& \hfill Y& ={M}_{\stackrel{\u0307}{2}\stackrel{\u0307}{2}},\phantom{\rule{2em}{0ex}}& \hfill Z& ={M}_{\stackrel{\u0307}{1}\stackrel{\u0307}{2}}={M}_{\stackrel{\u0307}{2}\stackrel{\u0307}{1}}.\phantom{\rule{2em}{0ex}}& \hfill \text{(7.3.28)}\end{array}$$The relations (7.3.24) and (7.3.25) give

$$AB={C}^{2},\phantom{\rule{2em}{0ex}}XY={Z}^{2},$$ | (7.3.29) |

whereas the relation (7.3.26) mean that two vectors $\left(A,B,C\right)$ and $\left(X,Y,Z\right)$ cannot be both nonzero at the same time.

Therefore we see that the Higgs branch has the structure schematically described in Fig. 7.2: there are two copies of ${\u2102}^{2}\u2215{\mathbb{Z}}_{2}$, described respectively by the variables $A,B,C$ and $X,Y,Z$. When the vacuum is on one of the ${\u2102}^{2}\u2215{\mathbb{Z}}_{2}$ described by one set of variables $\left(A,B,C\right)$, the other variables are forced to be zero, and vice versa. Therefore two copies of ${\u2102}^{2}\u2215{\mathbb{Z}}_{2}$ can be said to share the origin, where all of $A,B,C$ and $X,Y,Z$ are zero. The Higgs branch has complex dimension two, as expected.

Recall we decomposed the ﬂavor symmetry $SO\left(4\right)$ into $SU{\left(2\right)}_{u}\times SU{\left(2\right)}_{v}$. The vectors $\left(A,B,C\right)$ and $\left(X,Y,Z\right)$ are triplets under $SU{\left(2\right)}_{u}$ and $SU{\left(2\right)}_{v}$, respectively. Therefore, the ﬂavor parity of $O\left(4\right)\supset SO\left(4\right)$ exchanges the two copies of ${\u2102}^{2}\u2215{\mathbb{Z}}_{2}$ composing the Higgs branch.