Low energy Lagrangian

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

\begin{matrix} \leftrightarrow & λ_{α} & \leftrightarrow & A_{μ} & 𝒩 = 1 vector multiplet \\ a & \leftrightarrow & {\tilde{λ}}_{α} & \leftrightarrow & 𝒩 = 1 chiral multiplet \end{matrix}

(2.4.1)

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

\frac{1}{8 π} \int d^{4} 𝜃 K (ā_{i}, a_{j}) + \int d^{2} 𝜃 \frac{- i}{8 π} τ^{i j} (a) W_{α, i} W^{α}_{j} + c c .

(2.4.2)

Note that we allowed the Kähler potential and the gauge coupling matrix to have nontrivial dependence on $a_{i}$ .

We demand the existence of the $S U {(2)}_{R}$ symmetry rotating $λ_{α}$ and ${\tilde{λ}}_{α}$ to guarantee the existence of $𝒩 = 2$ supersymmetry. The kinetic matrix of ${\tilde{λ}}_{α}$ is

\frac{1}{4 π} \frac{\partial^{2} K}{\partial a_{i} \partial ā_{j}}

(2.4.3)

and that of $λ$ is

\frac{I m τ^{i j}}{2 π} = \frac{τ^{i j} - {\bar{τ}}^{i j}}{4 π i} .

(2.4.4)

Equating them, we have

\frac{τ^{i j} - {\bar{τ}}^{i j}}{i} = \frac{\partial^{2} K}{\partial a_{i} \partial ā_{j}} .

(2.4.5)

Taking the derivative of both sides by $a_{k}$ , we have

\frac{\partial}{\partial a_{k}} \frac{τ^{i j}}{i} = \frac{\partial^{3} K}{\partial a_{k} \partial a_{i} \partial ā_{j}} .

(2.4.6)

The left hand side is symmetric under $i \leftrightarrow j$ , and the right hand side is symmetric under $k \leftrightarrow i$ . Therefore, at least locally, $τ^{i j}$ can be integrated twice:

τ^{i j} = \frac{\partial^{2} F}{\partial a_{i} \partial a_{j}}

(2.4.7)

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

a_{D}^{i} = \frac{\partial F}{\partial a_{i}},

(2.4.8)

then we have

K = i (ā_{D}^{i} a_{i} - ā_{i} a_{D}^{i}) .

(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. [33, 34, 35] for a review. In this context, the ﬁelds $a_{i}$ and $a_{D}^{i}$ are called the special coordinates.

The notations $a_{i}$ and $a_{D}^{i}$ can be justiﬁed as follows. Suppose we have a hypermultiplet $Q$ , $\tilde{Q}$ charged under the $i$ -th vector multiplet only. It has the superpotential

W = Q a_{i} \tilde{Q},

(2.4.10)

which gives the mass

M_{Q} = | a_{i} | .

(2.4.11)

Therefore, $a_{i}$ is indeed the coeﬃcient appearing in (2.3.9). To justify the notation $a_{D}^{i}$ , write down the Lagrangian for the bosons in components:

\frac{I m τ^{i j}}{4 π} \partial_{μ} ā_{i} \partial^{μ} a_{j} + \frac{I m τ^{i j}}{8 π} F_{μ ν i} F_{j}^{μ ν} + \frac{R e τ^{i j}}{8 π} F_{μ ν i} {\tilde{F}}_{j}^{μ ν} .

(2.4.12)

Generalizing the argument in Sec. 1.2, the dual electromagnetic ﬁeld $F_{D}$ is given by

F_{D}_{μ ν}^{i} = I m τ^{i j} F_{μ ν j} + R e τ^{i j} {\tilde{F}}_{μ ν j},

(2.4.13)

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

\frac{1}{8 π} (I m τ_{D}_{i j} F_{D}_{μ ν}^{i} F_{D}^{μ ν j} + R e τ_{D}_{i j} F_{D}_{μ ν i} {\tilde{F}}_{D}^{μ ν j})

(2.4.14)

where

τ_{D}_{i j} = {(- τ^{- 1})}_{i j} .

(2.4.15)

Then we ﬁnd

\frac{1}{4 π} I m τ^{i j} \partial_{μ} ā_{i} \partial^{μ} a_{j} = \frac{1}{4 π} I m τ_{D}_{i j} \partial_{μ} ā_{D}^{i} \partial^{μ} a_{D}^{j}

(2.4.16)

where $a_{D}$ is as deﬁned in (2.4.8). This means that we have the dual $𝒩 = 2$ multiplets

\begin{matrix} \leftrightarrow & λ_{D}_{α} & \leftrightarrow & A_{D}_{μ} & 𝒩 = 1 vector multiplet \\ a_{D} & \leftrightarrow & {\tilde{λ}}_{D}_{α} & \leftrightarrow & 𝒩 = 1 chiral multiplet \end{matrix}

(2.4.17)

where $A_{D}_{μ}$ is the gauge potential of $F_{D}_{μ ν}$ 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

Φ_{i} = a_{i} + 2 {\tilde{λ}}_{i}^{α} 𝜃_{α} + \dots, W_{i} = λ_{α i} + F_{α β} 𝜃^{β} + \dots .

(2.4.18)

We can introduce another set of supercoordinates ${\tilde{𝜃}}_{α}$ to combine them:

Φ_{i} = Φ_{i} + 2 W_{α i} {\tilde{𝜃}}^{α} = a_{i} + 2 {\tilde{λ}}_{α i} 𝜃^{α} + 2 λ_{α i} {\tilde{𝜃}}^{α} + 2 F_{α β i} 𝜃^{(α} {\tilde{𝜃}}^{β)} + \dots .

(2.4.19)

Then the $S U (2)$ R-symmetry rotating $λ$ and $\tilde{λ}$ acts on the two sets of supercoordinates $𝜃_{α}$ and ${\tilde{𝜃}}_{α}$ .

Now, take an arbitrary holomorphic function of $n$ variables $F (a_{1}, \dots, a_{n})$ , and consider its integral over the chiral $𝒩 = 2$ superspace:

\int d^{2} 𝜃 d^{2} \tilde{𝜃} F (Φ_{1}, \dots, Φ_{n}) + c c .

(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 $F (Φ) = \frac{1}{2} τ t r Φ^{2}$ .

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