2.4 Low energy Lagrangian

Let us consider a general effective Lagrangian which describes U(1)n gauge fields in the infrared. Let us denote n U(1) vector multiplets by

λα Aμ𝒩=1 vector multiplet a λ̃α 𝒩=1 chiral multiplet (2.4.1)

with additional scripts i = 1,,n. A general 𝒩=1 supersymmetric Lagrangian is given by

1 8πd4𝜃K(āi,aj) +d2𝜃 i 8π τij(a)Wα,iWαj + cc. (2.4.2)

Note that we allowed the Kähler potential and the gauge coupling matrix to have nontrivial dependence on ai.

We demand the existence of the SU(2)R symmetry rotating λα and λ̃α to guarantee the existence of 𝒩=2 supersymmetry. The kinetic matrix of λ̃α is

1 4π 2K aiāj (2.4.3)

and that of λ is

Imτij 2π = τij τ̄ij 4πi . (2.4.4)

Equating them, we have

τij τ̄ij i = 2K aiāj. (2.4.5)

Taking the derivative of both sides by ak, we have

ak τij i = 3K akaiāj. (2.4.6)

The left hand side is symmetric under i j, and the right hand side is symmetric under k i. Therefore, at least locally, τij can be integrated twice:

τij = 2F aiaj (2.4.7)

for a locally holomorphic function F(a). We define

aDi = F ai, (2.4.8)

then we have

K = i(āDiai āiaDi). (2.4.9)

A Kähler manifold with this additional structure is often called a special Kähler manifold. With supergravity, a slightly different structure appears. To distinguish from it, it is also called a rigid special Kähler manifold. The same geometry is also called a Seiberg-Witten integrable system, or a Donagi-Witten integrable system. See e.g. [333435] for a review. In this context, the fields ai and aDi are called the special coordinates.

The notations ai and aDi can be justified as follows. Suppose we have a hypermultiplet Q, Q̃ charged under the i-th vector multiplet only. It has the superpotential

W = QaiQ̃, (2.4.10)

which gives the mass

MQ = |ai|. (2.4.11)

Therefore, ai is indeed the coefficient appearing in (2.3.9). To justify the notation aDi, write down the Lagrangian for the bosons in components:

Imτij 4π μāiμaj + Imτij 8π FμνiFjμν + Reτij 8π FμνiF̃jμν. (2.4.12)

Generalizing the argument in Sec. 1.2, the dual electromagnetic field FD is given by

FDμνi = ImτijFμνj + ReτijF̃μνj, (2.4.13)

in terms of which the kinetic term of the gauge fields is

1 8π ImτDijFDμνiFDμνj + ReτDijFDμνiF̃Dμνj (2.4.14)


τDij = (τ1)ij. (2.4.15)

Then we find

1 4πImτijμāiμaj = 1 4πImτDijμāDiμaDj (2.4.16)

where aD is as defined in (2.4.8). This means that we have the dual 𝒩=2 multiplets

λDα ADμ𝒩=1 vector multiplet aD λ̃Dα 𝒩=1 chiral multiplet (2.4.17)

where ADμ is the gauge potential of FDμν introduced above, with additional superscripts i.

We introduced the prepotential F in a rather indirect manner in this section, by saying that the kinetic term of the U(1) vector multiplets (2.4.2) should be given by (2.4.7) and (2.4.9). This can be better understood using 𝒩=2 superspace, since it is known that the prepotential is the Lagrangian density in the 𝒩=2 superspace. This is similar to the situation where the Kähler potential gives the Lagrangian density in the 𝒩=1 superspace.

Recall that the multiplets (2.4.1) can be summarized in 𝒩=1 superfields

Φi = ai + 2λ̃iα𝜃α + ,Wi = λαi + Fαβ𝜃β + . (2.4.18)

We can introduce another set of supercoordinates 𝜃̃α to combine them:

Φi = Φi + 2Wαi𝜃̃α = ai + 2λ̃αi𝜃α + 2λαi𝜃̃α + 2Fαβi𝜃(α𝜃̃β) + . (2.4.19)

Then the SU(2) R-symmetry rotating λ and λ̃ acts on the two sets of supercoordinates 𝜃α and 𝜃̃α.

Now, take an arbitrary holomorphic function of n variables F(a1,,an), and consider its integral over the chiral 𝒩=2 superspace:

d2𝜃d2𝜃̃F(Φ1,,Φn) + cc. (2.4.20)

It is clear that this gives rise to the structure (2.4.7) for the gauge kinetic matrix. To obtain the Kähler potential (2.4.9) one needs to study the structure of the constraints and the auxiliary fields of the 𝒩=2 superfields, see e.g. Sec. 2.10 of [14]. The non-Abelian microscopic action (2.1.11) has the prepotential F(Φ) = 1 2τtrΦ2.