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 SU(N) gauge theory with Nf 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 (Qi,Q̃i) in the representation R. Here the index i = 1,, dim R is for the hypermultiplet and we use the index a = 1,, dim G for the adjoint representation. The Higgs branch is given by

(QiQj Q̃iQ̃j)Taij = 0 ReQiQ̃jTaij = 0 ImQiQ̃jTaij = 0 /(identification by the gauge group) (7.3.1)

where Taij 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 dim G real constraints, for each I, J and K. Then we make the identification by the action of G. Therefore we have

4m 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

QiQ̃jTaij = 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 2m complex scalars. We then imposes dim G complex constraints and then perform the identification by the action of G, the complexified gauge group, removing dim G complex dimensions. We end up with

2m 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 (Qi,Q̃i) with charge ± 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

Q1Q̃1 + Q2Q̃2 = 0. (7.3.6)

Then we have

(Q1,Q̃1) = (z,z̃t),(Q2,Q̃2) = (z̃,zt) (7.3.7)

for some complex numbers z, z̃ and t. Then the D-term equation |Q1|2 + |Q2|2 |Q̃1|2 |Q̃2| = 0 says

|z|2 + |z̃|2 = |t|2(|z|2 + |z̃|2). (7.3.8)

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

Argz = Arg(z̃t). (7.3.9)

This still does not fix the U(1) gauge transformation given by the multiplication by 1 on Qi, Q̃i, sending the pair (z,z̃) to (z,z̃). We conclude that the Higgs branch is given by

22 = {(z,z̃) 2}(z,z̃) (z,z̃). (7.3.10)

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

Figure 7.1: Not a very accurate depiction of 22

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, Qi has charge + 1 and Q̃i has charge 1. Then, the gauge invariants are QiQ̃j, for arbitrary choices of i and j. We need to impose

Q1Q̃1 + Q2Q̃2 = 0, (7.3.11)

too. In total, we find three combinations

A = Q1Q̃2,B = Q2Q̃1,C = iQ1Q̃1 = iQ2Q̃2. (7.3.12)

They satisfy one obvious relation

AB = C2. (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 22 in (7.3.10). Combinations of z, z̃ invariant under (z,z̃) (z,z̃) are

A = z2,B = z̃2,C = zz̃ (7.3.14)

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

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

As the second example, consider 𝒩=2 SU(2) gauge theory with Nf full hypermultiplets in the doublet representations. In terms of 𝒩=1 chiral multiplets, we have

Qia,Q̃ai(a = 1, 2; i = 1,,Nf) (7.3.15)

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

QIa,(a = 1, 2; I = 1,, 2Nf) (7.3.16)

which makes SO(2Nf) flavor symmetry more manifest.

We have 4Nf complex scalars and dim SU(2) = 3. Then the complex dimension of the Higgs branch is

4Nf 2 3. (7.3.17)

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

Gauge-invariant combinations of QIa are

MIJ = QIaQJb𝜖ab. (7.3.18)

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

QI(aQJb)δIJ = 0. (7.3.19)

For SO(2Nf) = SO(4), we can split an antisymmetric matrix MIJ of SO(4) into the self-dual and the anti-self-dual parts. Equivalently, using SO(4) SU(2)u ×SU(2)v, MIJ splits into the triplet M(αβ) of SU(2)u and the triplet M(α̇β̇) of SU(2)v, where α,β = 1, 2 and α̇,β̇ = 1, 2 are doublet indices of SU(2)u,v respectively. The index I itself can be thought of a pair of indices: I = (αα̇). Then the hypermultiplets we are dealing with can be written as

Qaαα̇,a,α,α̇ = 1, 2 (7.3.20)

which makes the existence of SU(2)3 symmetry manifest. Then

Mαβ = Qaαα̇Qbββ̇𝜖ab𝜖α̇β̇, (7.3.21) Mα̇β̇ = Qaαα̇Qbββ̇𝜖ab𝜖αβ (7.3.22)

are the self-dual and the anti-self-dual parts of MIJ, respectively. The F-term equation can be written as

Qaαα̇Qbββ̇𝜖αβ𝜖α̇β̇ = 0. (7.3.23)

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

MαβMγδ𝜖αγ𝜖βδ = 0, (7.3.24) Mα̇β̇Mγ̇δ̇𝜖α̇γ̇𝜖β̇δ̇ = 0, (7.3.25) MαβMα̇β̇ = 0. (7.3.26)

The structure becomes clearer by defining

A = M11, B = M22, C = M12 = M21; (7.3.27) X = M1 ̇1 ̇, Y = M2 ̇2 ̇, Z = M1 ̇2 ̇ = M2 ̇1 ̇. (7.3.28)

The relations (7.3.24) and (7.3.25) give

AB = C2,XY = Z2, (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 22 22

Therefore we see that the Higgs branch has the structure schematically described in Fig. 7.2: there are two copies of 22, described respectively by the variables A,B,C and X,Y,Z. When the vacuum is on one of the 22 described by one set of variables (A,B,C), the other variables are forced to be zero, and vice versa. Therefore two copies of 22 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(4) into SU(2)u ×SU(2)v. The vectors (A,B,C) and (X,Y,Z) are triplets under SU(2)u and SU(2)v, respectively. Therefore, the flavor parity of O(4) SO(4) exchanges the two copies of 22 composing the Higgs branch.