The hyperkahler quotient

7.3 The hyperkähler quotient

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 $S U (N)$ 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 $𝒩 = 2$ gauge theory with gauge group $G$ and a hypermultiplet $(Q^{i}, {\tilde{Q}}_{i})$ in the representation $R$ . Here the index $i = 1, \dots, dim R$ is for the hypermultiplet and we use the index $a = 1, \dots, dim G$ for the adjoint representation. The Higgs branch is given by

\{\begin{matrix} (Q^{i} Q^{†}_{j} - {\tilde{Q}}^{†}^{i} {\tilde{Q}}_{j}) T^{a}_{i}^{j} = 0 \\ R e Q^{i} {\tilde{Q}}_{j} T^{a}_{i}^{j} = 0 \\ I m Q^{i} {\tilde{Q}}_{j} T^{a}_{i}^{j} = 0 \end{matrix}\} / (identiﬁcation by the gauge group)

(7.3.1)

where $T^{a}_{i}^{j}$ 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 $4 m$ real scalars. The D-term condition imposes $dim G$ real constraints, for each $I$ , $J$ and $K$ . Then we make the identiﬁcation by the action of $G$ . Therefore we have

4 m - 3 dim G - dim G = 4 (m - dim G)

(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

\{Q^{i} {\tilde{Q}}_{j} T^{a}_{i}^{j} = 0\} / (identiﬁcation by the complexiﬁed gauge group) .

(7.3.3)

Note that this is a more natural form in the $𝒩 = 1$ 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 $2 m$ complex scalars. We then imposes $dim G$ complex constraints and then perform the identiﬁcation by the action of $G_{ℂ}$ , the complexiﬁed gauge group, removing $dim G$ complex dimensions. We end up with

2 m - dim G - dim G = 2 (m - dim G)

(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 - dim G .

(7.3.5)

7.3.1 $U (1)$ gauge theory with one charged hypermultiplet

Let us consider two examples. First, take an $𝒩 = 2$ $U (1)$ gauge theory with two hypermultiplets $(Q_{i}, {\tilde{Q}}^{i})$ with charge $\pm 1$ . Here $i = 1, 2$ . We have $m = 2$ and $dim G = 1$ 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} {\tilde{Q}}^{1} + Q_{2} {\tilde{Q}}^{2} = 0 .

(7.3.6)

Then we have

(Q_{1}, {\tilde{Q}}^{1}) = (z, \tilde{z} t), (Q_{2}, {\tilde{Q}}^{2}) = (\tilde{z}, - z t)

(7.3.7)

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

| z |^{2} + | \tilde{z} |^{2} = | t |^{2} (| z |^{2} + | \tilde{z} |^{2}) .

(7.3.8)

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

A r g z = A r g (\tilde{z} t) .

(7.3.9)

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

ℂ^{2} ∕ ℤ_{2} = {(z, \tilde{z}) \in ℂ^{2}} ∕ (z, \tilde{z}) \leftrightarrow (- z, - \tilde{z}) .

(7.3.10)

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

Figure 7.1: Not a very accurate depiction of

ℂ^{2} ∕ ℤ_{2}

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 ${\tilde{Q}}^{i}$ has charge $- 1$ . Then, the gauge invariants are $Q_{i} {\tilde{Q}}^{j}$ , for arbitrary choices of $i$ and $j$ . We need to impose

Q_{1} {\tilde{Q}}^{1} + Q_{2} {\tilde{Q}}^{2} = 0,

(7.3.11)

too. In total, we ﬁnd three combinations

A = Q_{1} {\tilde{Q}}^{2}, B = Q_{2} {\tilde{Q}}^{1}, C = i Q_{1} {\tilde{Q}}^{1} = - i Q_{2} {\tilde{Q}}^{2} .

(7.3.12)

They satisfy one obvious relation

A B = 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 $ℂ^{2} ∕ ℤ_{2}$ in (7.3.10). Combinations of $z$ , $\tilde{z}$ invariant under $(z, \tilde{z}) \leftrightarrow (- z, - \tilde{z})$ are

A = z^{2}, B = {\tilde{z}}^{2}, C = z \tilde{z}

(7.3.14)

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

7.3.2 $S U (2)$ gauge theory with two hypermultiplets in the doublet

As the second example, consider $𝒩 = 2$ $S U (2)$ gauge theory with $N_{f}$ full hypermultiplets in the doublet representations. In terms of $𝒩 = 1$ chiral multiplets, we have

Q_{i}^{a}, {\tilde{Q}}_{a}^{i} (a = 1, 2; i = 1, \dots, N_{f})

(7.3.15)

As the doublet and the anti-doublet representations are the same for $S U (2)$ gauge theory, we can denote them also as

Q_{I}^{a}, (a = 1, 2; I = 1, \dots, 2 N_{f})

(7.3.16)

which makes $S O (2 N_{f})$ ﬂavor symmetry more manifest.

We have $4 N_{f}$ complex scalars and $dim S U (2) = 3$ . 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_{I J} = Q_{I}^{a} Q_{J}^{b} 𝜖_{a b} .

(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} δ^{I J} = 0 .

(7.3.19)

For $S O (2 N_{f}) = S O (4)$ , we can split an antisymmetric matrix $M_{I J}$ of $S O (4)$ into the self-dual and the anti-self-dual parts. Equivalently, using $S O (4) ≃ S U {(2)}_{u} \times S U {(2)}_{v}$ , $M_{I J}$ splits into the triplet $M_{(α β)}$ of $S U {(2)}_{u}$ and the triplet $M_{(\dot{α} \dot{β})}$ of $S U {(2)}_{v}$ , where $α, β = 1, 2$ and $\dot{α}, \dot{β} = 1, 2$ are doublet indices of $S U {(2)}_{u, v}$ respectively. The index $I$ itself can be thought of a pair of indices: $I = (α \dot{α})$ . Then the hypermultiplets we are dealing with can be written as

Q_{a α \dot{α}}, a, α, \dot{α} = 1, 2

(7.3.20)

which makes the existence of $S U {(2)}^{3}$ symmetry manifest. Then

\begin{align} M_{α β} & = Q_{a α \dot{α}} Q_{b β \dot{β}} 𝜖^{a b} 𝜖^{\dot{α} \dot{β}}, & (7.3.21) \\ M_{\dot{α} \dot{β}} & = Q_{a α \dot{α}} Q_{b β \dot{β}} 𝜖^{a b} 𝜖^{α β} & (7.3.22) \end{align}

are the self-dual and the anti-self-dual parts of $M_{I J}$ , respectively. The F-term equation can be written as

Q_{a α \dot{α}} Q_{b β \dot{β}} 𝜖^{α β} 𝜖^{\dot{α} \dot{β}} = 0 .

(7.3.23)

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

\begin{align} M_{α β} M_{γ δ} 𝜖^{α γ} 𝜖^{β δ} & = 0, & (7.3.24) \\ M_{\dot{α} \dot{β}} M_{\dot{γ} \dot{δ}} 𝜖^{\dot{α} \dot{γ}} 𝜖^{\dot{β} \dot{δ}} & = 0, & (7.3.25) \\ M_{α β} M_{\dot{α} \dot{β}} & = 0 . & (7.3.26) \end{align}

The structure becomes clearer by deﬁning

\begin{align} A & = M_{11}, & B & = M_{22}, & C & = M_{12} = M_{21}; & (7.3.27) \\ X & = M_{\dot{1} \dot{1}}, & Y & = M_{\dot{2} \dot{2}}, & Z & = M_{\dot{1} \dot{2}} = M_{\dot{2} \dot{1}} . & (7.3.28) \end{align}

The relations (7.3.24) and (7.3.25) give

A B = C^{2}, X Y = Z^{2},

(7.3.29)

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

Figure 7.2: Not a very accurate depiction of

ℂ^{2} ∕ ℤ_{2} \land ℂ^{2} ∕ ℤ_{2}

Therefore we see that the Higgs branch has the structure schematically described in Fig. 7.2: there are two copies of $ℂ^{2} ∕ ℤ_{2}$ , described respectively by the variables $A, B, C$ and $X, Y, Z$ . When the vacuum is on one of the $ℂ^{2} ∕ ℤ_{2}$ described by one set of variables $(A, B, C)$ , the other variables are forced to be zero, and vice versa. Therefore two copies of $ℂ^{2} ∕ ℤ_{2}$ 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 $S O (4)$ into $S U {(2)}_{u} \times S U {(2)}_{v}$ . The vectors $(A, B, C)$ and $(X, Y, Z)$ are triplets under $S U {(2)}_{u}$ and $S U {(2)}_{v}$ , respectively. Therefore, the ﬂavor parity of $O (4) \supset S O (4)$ exchanges the two copies of $ℂ^{2} ∕ ℤ_{2}$ composing the Higgs branch.

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7.3 The hyperkähler quotient

7.3.1 U(1) gauge theory with one charged hypermultiplet

7.3.2 SU(2) gauge theory with two hypermultiplets in the doublet

7.3.1 $U (1)$ gauge theory with one charged hypermultiplet

7.3.2 $S U (2)$ gauge theory with two hypermultiplets in the doublet