Why a holomorphic prepotential controls the couplings

A photon kinetic term measures a gauge coupling. A scalar kinetic term measures the cost of moving through the space of vacua. In an ordinary N=1 theory these can be chosen independently. Why does the Seiberg–Witten solution determine both from one complex function?

Read running couplings and the scale Lambda and N=2 vectors and hypermultiplets first. We retain pure four-dimensional N=2 $SU(2)$ theory: eight real supercharges, with four manifest in N=1 superspace. At a regular Coulomb-branch vacuum only a rank-one Abelian vector multiplet remains light. We work locally, at energies well below the lightest omitted charged mass, and keep the terms with at most two spacetime derivatives.

Integrating out a field means performing its functional integral while keeping the light fields as arguments of the resulting action. A massive propagator illustrates the local expansion: at momenta small compared with its mass, $(m^2-p^2)^{-1}=m^{-2}+p^2m^{-4}+\cdots$. Heavy-field diagrams similarly give local terms ordered by derivatives; their coefficients depend on the chosen vacuum and matching prescription. The omitted terms need not vanish, but they are suppressed in this regime. If a charged state becomes light, this expansion fails and that state must be retained.

The multiplet consists of a neutral chiral superfield $A$, with components $(a,\psi,F_A)$, and an Abelian vector superfield, with components $(v_\mu,\lambda,D)$. The scalar $a$ is the first paper’s electric period; it has mass dimension one. The auxiliary $F_A$ is a field. The calligraphic symbol $\mathcal F$ below is a function.

One action, two apparent choices

An N=1 description could start with a real Kähler potential $K(A,\bar A)$ and a holomorphic gauge function $f_{\rm g}(A)$:

$$\mathcal L=\int d^4\theta\,K +\left[\frac14\int d^2\theta\,f_{\rm g}(A)W^\alpha W_\alpha +\mathrm{h.c.}\right]. \tag{SW05.1}$$

Here $W_\alpha$ is the chiral field-strength superfield, not a superpotential. “Holomorphic” means dependence on $A$ without dependence on $\bar A$. The coefficients of the two fermion kinetic terms are respectively $K_{a\bar a}$ and $\operatorname{Re}f_{\rm g}$. In the normalization where $\psi$ and $\lambda$ form the N=2 R-symmetry doublet, they must agree. Write

$$f_{\rm g}=-\frac{i}{4\pi}\tau(a),\qquad K_{a\bar a}=\frac{\operatorname{Im}\tau(a)}{4\pi}. \tag{SW05.2}$$

This equality explains the physical restriction. By itself a match of quadratic terms is not a proof of the entire second supersymmetry. The full N=2 completion is the structural input: its two-derivative action has the form in Seiberg–Witten, (2.7). Tachikawa, (2.4.2)–(2.4.9) explains the fermion-kinetic argument in his normalization.

On a simply connected small patch a holomorphic $\tau$ can be integrated twice. Define the prepotential $\mathcal F$, of mass dimension two, by

$$a_D(a)=\mathcal F'(a),\qquad \tau(a)=\mathcal F''(a). \tag{SW05.3}$$

A prime denotes differentiation with respect to the local coordinate $a$. A real function with the required mixed derivative is

$$K_{\rm phys} =\frac{\bar a\,\mathcal F'(a)-a\,\overline{\mathcal F'(a)}}{8\pi i} =\frac{1}{4\pi}\operatorname{Im}(\bar a a_D). \tag{SW05.4}$$

Indeed, differentiating first in $a$ and then in $\bar a$ gives

$$\partial_a K_{\rm phys} =\frac{\bar a\,\tau-\bar a_D}{8\pi i}, \qquad G\equiv\partial_{\bar a}\partial_aK_{\rm phys} =\frac{\tau-\bar\tau}{8\pi i} =\frac{\operatorname{Im}\tau}{4\pi}. \tag{SW05.5}$$

Any other real $K$ with this mixed derivative differs locally by a holomorphic function plus its complex conjugate. That change is a Kähler transformation and contributes no bulk D-term. The subscript “phys” keeps the overall $1/(4\pi)$ inside $K$; the metric written in the first paper often leaves this common action factor outside.

Combining (SW05.1)–(SW05.4) gives

$$\boxed{\displaystyle \mathcal L_2=\frac{1}{4\pi}\operatorname{Im} \left[\int d^4\theta\,\bar A\,\mathcal F'(A) +\frac12\int d^2\theta\,\mathcal F''(A)W^\alpha W_\alpha\right].} \tag{SW05.6}$$

The imaginary part includes the conjugate action; it is not an instruction to discard the fields’ complex dependence. Equation (SW05.6) packages the fermionic interactions as well as the bosonic terms we now extract.

Extract the scalar and photon terms

Use Lorentzian signature $(+---)$ and the inherited chiral expansion $A=a+\sqrt2\theta\psi+\theta^2F_A$ in chiral coordinates. To see the scalar derivative coefficient explicitly, set fermions to zero and put $B(A)=\mathcal F'(A)$. Its lowest and auxiliary components are $B(a)$ and $B'(a)F_A$. The top component of $\bar A B(A)$, before integrating by parts, is

$$-\frac14\bar a\,\Box B-\frac14B\,\Box\bar a +\frac12\partial_\mu\bar a\,\partial^\mu B +B'\bar F_AF_A. \tag{SW05.7}$$

Each of the first two terms becomes one quarter of $\partial_\mu\bar a\,\partial^\mu B$, up to a divergence. Therefore their sum with the third term is $B'\partial_\mu\bar a\,\partial^\mu a$. Since $B'=\tau$, taking the imaginary part yields the scalar and chiral-auxiliary coefficient $G$ in (SW05.5). This calculation also explains why the derivative acts on the whole composite $B(a)$.

For the vector define $f_{\mu\nu}=\partial_\mu v_\nu-\partial_\nu v_\mu$ and $\widetilde f^{\mu\nu}=\tfrac12\epsilon^{\mu\nu\rho\sigma}f_{\rho\sigma}$, with $\epsilon_{0123}=+1$, hence $\epsilon^{0123}=-1$ in signature $(+---)$. The inherited Weyl matrices are $\sigma^\mu=(1,-\boldsymbol\sigma)$ and $\bar\sigma^\mu=(1,+\boldsymbol\sigma)$. The bosonic chiral top component is

$$\left.W^\alpha W_\alpha\right|_{\theta^2,\mathrm{bos}} =D^2-\frac12 f_{\mu\nu}f^{\mu\nu} +\frac{i}{2}f_{\mu\nu}\widetilde f^{\mu\nu}. \tag{SW05.8}$$

For example, write $W_\alpha=\theta_\alpha D+i(\sigma^{\mu\nu}\theta)_\alpha f_{\mu\nu}$ in the bosonic sector, where $\sigma^{\mu\nu}=(\sigma^\mu\bar\sigma^\nu-\sigma^\nu\bar\sigma^\mu)/4$. The traceless matrix multiplying $\theta$ removes the cross term linear in $D$; its quadratic contraction gives the last two terms. A direct check sets $f_{03}=E$ and $f_{12}=B$: the matrix multiplying $\theta$ is $D\,1+(B+iE)\sigma_3$, whose determinant is $D^2+E^2-B^2-2iEB$. Our orientation gives $f\widetilde f=-4EB$, in agreement with (SW05.8). This translates Bilal’s component formulas (4.46)–(4.50): the upper-index epsilon sign printed there is inconsistent with his explicit Weyl matrices, so we fix the orientation by this calculation.

Substitution in (SW05.6) gives the complete bosonic two-derivative density, up to a divergence:

$$\boxed{\displaystyle \mathcal L_{2,\mathrm{bos}} =G\,\partial_\mu\bar a\,\partial^\mu a -\frac{\operatorname{Im}\tau}{16\pi}f_{\mu\nu}f^{\mu\nu} +\frac{\operatorname{Re}\tau}{16\pi}f_{\mu\nu}\widetilde f^{\mu\nu} +G\left(\bar F_AF_A+\frac12D^2\right).} \tag{SW05.9}$$

In particular,

$$\frac1{g_{\rm eff}^2}=\frac{\operatorname{Im}\tau}{4\pi}=G, \qquad \theta_{\rm eff}=2\pi\operatorname{Re}\tau. \tag{SW05.10}$$

Both scalar and photon have positive kinetic energy precisely when $\operatorname{Im}\tau>0$. A constant theta term is a local divergence, but its global quantum effect need not vanish. When $\tau$ varies with $a$, even its local variation couples the scalar to the photon.

Check the normalization at tree level. With $\mathcal F=\tau_{\rm cl}a^2/2$ and $\operatorname{Im}\tau_{\rm cl}=4\pi/g^2$, take the canonical microscopic fields $\phi=b\,t^3$, $t^3=\sigma_3/2$, $\operatorname{Tr}_2(t^3t^3)=1/2$. Then $a=gb$ and $v_\mu=gA_\mu^3$ for constant microscopic $g$. The first two terms of (SW05.9) become $|\partial b|^2-\tfrac14(F_{\mu\nu}^3)^2$, exactly their canonical normalization. There is no extra fundamental trace of $1/2$ to insert into the already component-valued Abelian action. Nor can one replace $g$ by $g_{\rm eff}(a)$ in this field redefinition without generating derivative terms.

On a bosonic configuration without superpotential, charged light matter or an FI deformation, the auxiliary equations give $F_A=D=0$. Thus a nonconstant prepotential changes kinetic terms and interactions but creates no scalar potential. The auxiliary equations acquire fermion bilinears when fermions are restored; those terms belong to the same supersymmetric completion.

A local deformation you can calculate

Consider the illustrative local effective theory

$$\mathcal F_{\rm toy}(a) =\frac{iT}{2}a^2+\frac{i\varepsilon}{12M^2}a^4, \qquad T>0,\quad M>0,\quad\varepsilon\in\mathbb R. \tag{SW05.11}$$

Here $T,\varepsilon$ are dimensionless. This is a test prepotential, not a proposed exact solution of pure $SU(2)$. Its higher-dimensional coupling is useful on a specified effective-theory patch. Differentiating gives

$$a_D=iTa+\frac{i\varepsilon}{3M^2}a^3,\qquad \tau=iT+\frac{i\varepsilon}{M^2}a^2. \tag{SW05.12}$$

Write $a=X+iY$, with real $X,Y$. The observable local coefficients are

$$\operatorname{Im}\tau=T+\frac{\varepsilon}{M^2}(X^2-Y^2), \qquad \operatorname{Re}\tau=-\frac{2\varepsilon XY}{M^2}, \qquad K_{\rm phys}=\frac1{4\pi} \left[T(X^2+Y^2)+\frac{\varepsilon}{3M^2}(X^4-Y^4)\right]. \tag{SW05.13}$$

The last expression is independently checked by applying $\partial_a\partial_{\bar a}=(\partial_X^2+\partial_Y^2)/4$. If $|a|\le M/2$, then $\operatorname{Im}\tau\ge T-|\varepsilon|/4$. Taking $T>|\varepsilon|/4$ therefore guarantees positive kinetic energy throughout that disk. Extending the polynomial to a region where its kinetic coefficient vanishes would not define a healthy continuation of this effective theory.

For $T=4$, $\varepsilon=1$ and $a_0=M(1+i)/4$,

$$\tau(a_0)=-\frac18+4i,\qquad G(a_0)=\frac1\pi,\qquad g_{\rm eff}^2(a_0)=\pi,\qquad \theta_{\rm eff}(a_0)=-\frac{\pi}{4}. \tag{SW05.14}$$

But $a_D(a_0)/a_0=-1/24+4i$. The ratio of periods is not the gauge coupling; the derivative ratio is. Around this vacuum, $\delta\tau=\mathcal F'''(a_0)\delta a+\cdots$ with $\mathcal F'''(a_0)=2i\varepsilon a_0/M^2$. Consequently the gauge density contains $-\operatorname{Im}(\mathcal F'''\delta a)f^2/(16\pi) +\operatorname{Re}(\mathcal F'''\delta a)f\widetilde f/(16\pi)$. This is an explicit scalar–photon interaction even though every constant $a$ in the patch still has zero potential.

For the actual weakly coupled pure theory, the preceding lesson instead gave $\mathcal F_{\rm pert}=ia^2\log(a^2/\Lambda_{\mathcal F}^2)/(2\pi)$. Our extraction then gives $G=[\log(|a|^2/|\Lambda_{\mathcal F}|^2)+3]/(4\pi^2)$ on its chosen logarithmic patch. The $+3$ and the finite definition of $\Lambda_{\mathcal F}$ remain those of the preceding lesson. Extrapolating this weak-coupling expression to small $|a|$ is not a determination of the strong-coupling metric.

What this function does and does not determine

The prepotential has dimension two and enters both a D-term and a gauge F-term in (SW05.6). An N=1 superpotential $\mathscr W$ has dimension three and enters $\int d^2\theta\,\mathscr W(A)$. If such a deformation is allowed and added, eliminating the chiral auxiliary gives $V=G^{-1}|\mathscr W'(a)|^2$ in the bosonic neutral theory. It generally changes the supersymmetry and the vacua. One must not insert $\mathcal F$ into that potential formula.

The local prepotential specifies the Wilsonian two-derivative action after heavy fields are removed, in the stated supersymmetric normalization. The full one-particle-irreducible action also includes propagation of the retained massless fields and can contain nonlocal momentum dependence. Higher-derivative local terms are additional data. Even an exact $\mathcal F$ would therefore not be a formula for every correlator or scattering amplitude.

Finally, $a$ is a special coordinate: it is paired by N=2 supersymmetry with the photon in a chosen electric normalization. Scalar-coordinate covariance still holds, but an arbitrary holomorphic relabeling does not preserve the simple operation “differentiate the composed prepotential twice.” Exercise 2 gives a sharp counterexample. The resulting metric structure is called rigid special Kähler geometry; here its meaning is the concrete relation (SW05.3)–(SW05.5), with no coupling to dynamical gravity.

For comparison with Tachikawa’s pure-$SU(2)$ period basis, $a_{\rm T}=a/\sqrt2$ and $a_{D,\rm T}=\sqrt2a_D$. Thus $\mathcal F_{\rm T}(a_{\rm T})=\mathcal F(\sqrt2a_{\rm T})$ implies $\tau_{\rm T}=2\tau$ and $\operatorname{Im}\tau_{\rm T}|da_{\rm T}|^2 =\operatorname{Im}\tau|da|^2$. His displayed component normalization must be translated together with the electromagnetic fields; it is not a reason to insert a factor of two into (SW05.9).

Exercises

1. Which changes leave the local action unchanged?

Let $\mathcal F\mapsto\mathcal F+c_0+c_1a+\tfrac12c_2a^2$. Find the changes in $a_D$, $\tau$, $K_{\rm phys}$ and the bosonic action. Separate a real $c_2$ from a complex one. Explain why the freedom in $c_1$ does not make absolute magnetic central charges arbitrary in a fixed microscopic theory.

Hint 1

Differentiate before interpreting the change. The constant $c_0$ has no derivative.

Hint 2

The term $\operatorname{Im}(c_1\bar a)$ has zero mixed derivative. A real $c_2$ changes the theta coefficient but not the metric.

Solution

The changes are

$$\Delta a_D=c_1+c_2a,\qquad \Delta\tau=c_2,\qquad \Delta K_{\rm phys} =\frac{\operatorname{Im}(c_1\bar a)}{4\pi} +\frac{\operatorname{Im}c_2}{4\pi}|a|^2.$$

The first term is holomorphic plus antiholomorphic and contributes only a D-term boundary term. Thus $c_0,c_1$ leave the local bulk action unchanged. If $c_2$ is real, only the topological term changes: $\Delta\mathcal L=c_2 f\widetilde f/(16\pi)$ and $\Delta\theta=2\pi c_2$. A local divergence is not automatically globally unobservable. Periodicity and allowed changes of electromagnetic basis require the theory’s charge and global data. If $\operatorname{Im}c_2\ne0$, then $\Delta G=\operatorname{Im}c_2/(4\pi)$ and the scalar, photon and auxiliary kinetic coefficients change as in (SW05.9). Positivity must be checked again.

Although the bulk action does not determine $c_1$, the physical period $a_D$ enters $Z_\gamma=n_ma_D+n_ea$. Its additive constant is fixed by matching to the microscopic theory and its chosen period conventions. Treating that matching as optional would change magnetic BPS masses while pretending to keep the same theory. This is the qualification following the first paper’s (2.13).

2. A holomorphic coordinate trap

Start from $\mathcal F=iTa^2/2$, $T>0$, and set $w=a^2/M$ on a patch with $a\ne0$ and a fixed square-root branch. Compute the physical scalar metric in $w$. Compare it with the second derivative of the composed function $\mathcal F(a(w))$. Has the photon lost its kinetic term?

Hint 1

Transform $G\,da\,d\bar a$ using $a=\sqrt{Mw}$.

Hint 2

For a nonlinear $a(w)$, the second derivative of a composition contains both $\mathcal F''(da/dw)^2$ and $\mathcal F'(d^2a/dw^2)$.

Solution

Since $da/dw=M/(2a)$, the metric becomes

$$G_w=\frac{T}{4\pi}\left|\frac{da}{dw}\right|^2 =\frac{TM}{16\pi|w|}>0.$$

One can also use $K_{\rm phys}=T|a|^2/(4\pi)=TM|w|/(4\pi)$. Away from $w=0$, its mixed derivative is $TM/(16\pi|w|)$. But the composed holomorphic function is $iTMw/2$, whose second derivative vanishes. The two chain-rule terms cancel:

$$(iT)\frac{M^2}{4a^2} +(iTa)\left(-\frac{M^2}{4a^3}\right)=0.$$

The photon still has $\tau=iT$ and a positive kinetic term. Only the scalar coordinate was relabeled; the photon was not changed into a multiplet for which $w$ is a special coordinate. In $w$ variables the action remains supersymmetric, but it is not obtained by that naive composition in (SW05.6). The chart is also invalid at $a=0$, so its metric singularity there is not evidence of a singularity in the original constant metric.

3. A local exchange and a positivity test

On a patch with $\operatorname{Im}\tau>0$, define $b=a_D$ and $b_D=-a$. Find $\tau_b=db_D/db$ and show that the scalar kinetic term is unchanged. Evaluate the transformation at the toy vacuum (SW05.14). Does this establish a global electric–magnetic duality or a complete particle spectrum?

Hint 1

The inverse-function theorem applies because a number in the upper half-plane is nonzero.

Hint 2

Use $db=\tau\,da$ and multiply $-1/\tau$ by $\bar\tau/\bar\tau$.

Solution

Locally $a$ is a holomorphic function of $b$, and

$$\tau_b=-\frac1{\tau},\qquad \operatorname{Im}\tau_b=\frac{\operatorname{Im}\tau}{|\tau|^2}>0, \qquad \frac{\operatorname{Im}\tau_b}{4\pi}|db|^2 =\frac{\operatorname{Im}\tau}{4\pi}|da|^2.$$

At $\tau=(-1+32i)/8$, $|\tau|^2=1025/64$ and $\tau_b=(8+256i)/1025$. Multiplying its imaginary part $256/1025$ by $1025/64$ restores $4$, the original coefficient before $1/(4\pi)$. The Kähler expression is unchanged as well: $\operatorname{Im}(\bar b b_D) =\operatorname{Im}(-\bar a_Da) =\operatorname{Im}(\bar a a_D)$.

This checks the local scalar geometry and its positivity under the exchange of period coordinates. It does not derive electromagnetic path-integral dualization, specify which period changes are globally allowed, or establish that a state of every possible charge exists. Those questions require the gauge field, charge lattice and continuation between patches.

The useful checkpoint is now concrete: given a local holomorphic $\mathcal F$, calculate $a_D$, $\tau$, $G$, the bosonic action and its positive domain. Electric–magnetic duality as a change of variables explains how different local descriptions fit together when no single positive coordinate patch covers the whole Coulomb branch.

Paper guides

Connect this lesson to the original papers and their notation.

• Reading the first Seiberg–Witten paper