Prepotentials and Special Kähler Geometry

A holomorphic prepotential packages the local two-derivative action of abelian $\mathcal N=2$ vector multiplets. Its Hessian is the complexified gauge coupling, and its first derivatives join the special coordinates into a symplectic period vector. The prepotential depends on a local electric polarization; the resulting rigid special Kähler geometry is the frame-independent structure.

Required background. The abelian Coulomb-branch theory fixes the fields, while holomorphic couplings and background superfields fixes what holomorphy controls. Helpful background. Moment maps and symplectic reduction supplies the symplectic vocabulary.

From the prepotential to the metric

On a rank-$r$ patch choose complex special coordinates $a^I$. Let

$$a_{D,I}=\partial_I\mathcal F(a), \qquad \tau_{IJ}=\partial_I\partial_J\mathcal F(a).$$

Holomorphy makes $\tau_{IJ}$ holomorphic, and equality of mixed derivatives makes it symmetric. A Kähler potential is

$$K(a,\bar a) =\operatorname{Im}\!\left(\bar a^Ia_{D,I}\right) =\frac{1}{2i} \left(\bar a^Ia_{D,I}-a^I\bar a_{D,I}\right).$$

Differentiating gives

$$g_{I\bar J} =\partial_I\partial_{\bar J}K =\operatorname{Im}\tau_{IJ}.$$

Therefore the scalar kinetic term and Maxwell kinetic matrix are the same positive matrix, as required by $\mathcal N=2$ supersymmetry. Physical positivity requires

$$v^I\operatorname{Im}\tau_{IJ}\bar v^J>0$$

for nonzero $v$ in the local effective theory.

Adding a quadratic polynomial to $\mathcal F$ changes theta angles and the symplectic basis according to its quantized part; adding a constant or linear term does not change $\tau$. The induced change in $K$ is a Kähler transformation or an irrelevant constant when the corresponding basis change is physical.

The symplectic section

Define

$$\Pi=\binom{a_D}{a}.$$

With

$$\Omega= \begin{pmatrix} 0&-\mathbf1\\ \mathbf1&0 \end{pmatrix},$$

the Kähler potential can be written

$$K=\frac{1}{2i}\,\overline\Pi^{,T}\Omega\Pi$$

in the displayed ordering. An integral frame change

$$\Pi' =M\Pi, \qquad M^T\Omega M=\Omega, \qquad M\in Sp(2r,\mathbb Z),$$

leaves this expression invariant. The integrality preserves the charge lattice; a general real symplectic matrix is a classical field redefinition but need not be a quantum duality frame.

The central charge is the symplectic pairing of a charge with $\Pi$. Depending on whether charge columns are ordered as $(p,q)$ or $(q,p)$, an explicit sign or transpose moves between formulas. State one ordering and transform charges contragrediently.

How the coupling transforms

Write the symplectic matrix in blocks,

$$M=\begin{pmatrix}A&B\\C&D\end{pmatrix}.$$

Since

$$\binom{da_D'}{da'} =M\binom{da_D}{da},$$

and $da_D=\tau\,da$, one obtains

$$\tau'=(A\tau+B)(C\tau+D)^{-1}$$

for the present ordering of $(a_D,a)$. If the period vector is ordered $(a,a_D)$, the familiar block formula is correspondingly permuted. This is why an isolated transformation rule for $\tau$ is unsafe without the period convention.

Symplectic identities and symmetry of $\tau$ ensure that $\tau'$ is symmetric. Its imaginary part transforms as

$$\operatorname{Im}\tau' =(C\bar\tau+D)^{-T} \operatorname{Im}\tau (C\tau+D)^{-1},$$

so positivity is preserved wherever the denominator is invertible.

Local existence of a new prepotential

After a frame change, $a'$ can serve as coordinates only if the Jacobian

$$\frac{\partial a'}{\partial a}=C\tau+D$$

is nonsingular. On such a patch, symmetry of $\tau'$ provides the integrability needed for a local $\mathcal F'(a')$. If the Jacobian degenerates, that electric polarization is not a valid coordinate chart there even though the symplectic section remains meaningful.

Monodromy can return $\Pi$ transformed after a loop, preventing one single-valued global $\mathcal F$. Rigid special Kähler geometry is therefore a flat symplectic local system plus a holomorphic section satisfying the special integrability and positivity conditions, not a globally chosen function. Mathematical definitions and their field-theory realization are developed in Freed 1999, §§1 and 5.

Special coordinates versus gauge invariants

Let $u^k$ be arbitrary holomorphic coordinates on the Coulomb branch. Pulling back the metric gives

$$g_{k\bar\ell} =\operatorname{Im}\tau_{IJ} \frac{\partial a^I}{\partial u^k} \frac{\partial\bar a^J}{\partial\bar u^{\bar\ell}}.$$

For rank one,

$$ds^2=\operatorname{Im}\tau(u) \left|\frac{da}{du}\right|^2du,d\bar u.$$

The gauge-invariant coordinate $u$ can be regular where $a$ has branch monodromy. Conversely, $da/du$ can vanish or diverge because the special coordinate is a poor chart. Metric singularities must be interpreted together with the light spectrum.

Perturbative and instanton terms

Dimensional analysis allows the asymptotically free rank-one form

$$\mathcal F(a) =\frac{i}{2\pi}a^2\log\frac{a^2}{\Lambda^2} +a^2\sum_{k\ge1}c_k \left(\frac{\Lambda}{a}\right)^{4k},$$

up to a quadratic polynomial and normalization choices. The first term is the one-loop threshold. The power series is generated by instanton sectors. Holomorphy and symmetry constrain the form but do not determine all $c_k$ without dynamical input.

Seiberg–Witten periods resum these contributions globally. Expanding them at large $a$ recovers the same series and supplies an independent normalization check; see the low-energy action and exact period solution in Seiberg and Witten 1994, §2.3 and §6.

What the prepotential omits

The local $\mathcal F$ does not by itself determine:

• higher-derivative vector interactions;
• hypermultiplet metrics;
• which BPS charges are populated in a chamber;
• global form and the set of genuine lines;
• a local Lagrangian at mutually nonlocal singularities;
• boundary or defect couplings.

Each omission has its own page or requires additional protected data.

Common pitfalls

Demanding one global prepotential. Monodromy can mix electric and magnetic variables, so only the symplectic section is globally patched.

Using a nonintegral symplectic transformation quantum mechanically. It can preserve the classical metric while destroying charge quantization.

Ignoring positivity. A holomorphic symmetric matrix is not automatically a physical gauge coupling; $\operatorname{Im}\tau$ must be positive on the patch.

Exercises

Take rank one with

$$\mathcal F(a)=\frac12\tau_0a^2, \qquad \operatorname{Im}\tau_0>0.$$

1. Compute $a_D$, $K$, and the metric.
2. Apply the $S$ transformation $(a_D',a')=(a,-a_D)$ and find $\tau'$.

Solution

$a_D=\tau_0a$ and

$$K=\operatorname{Im}(\tau_0)|a|^2, \qquad g_{a\bar a}=\operatorname{Im}\tau_0.$$

Since $a'= -a_D=-\tau_0a$ and $a_D'=a$, one finds

$$\tau'=\frac{da_D'}{da'}=-\frac1{\tau_0}.$$

Its imaginary part is $\operatorname{Im}\tau_0/|\tau_0|^2>0$, confirming positivity in the dual frame.

References

• Freed, Daniel S. “Special Kähler Manifolds.” Communications in Mathematical Physics 203 (1999): 31–52. arXiv:hep-th/9712042.
• Seiberg, Nathan, and Edward Witten. “Electric–Magnetic Duality, Monopole Condensation, and Confinement in $\mathcal N=2$ Supersymmetric Yang–Mills Theory.” Nuclear Physics B 426 (1994): 19–52; erratum 430 (1994): 485–486. arXiv:hep-th/9407087.