What does the Seiberg–Witten solution solve?

An exact solution of an interacting four-dimensional gauge theory sounds like a promise to calculate every scattering process. The Seiberg–Witten solution makes a more specific and remarkable prediction: it determines the leading low-energy dynamics along a family of supersymmetric vacua, including regions where the original gauge variables are strongly coupled.

Before introducing the geometry used to obtain that answer, we need a definite theory and a definite low-energy question. We will write its action, find a mass threshold in one simple background, and use that threshold to decide which questions a reduced description can answer.

The preparation is N=2 vector multiplets and hypermultiplets. Everything here is four-dimensional. The three-dimensional spinor and curved-background lessons are not required.

Define the microscopic theory

Work in flat Lorentzian spacetime with signature $(+---)$ and global $\mathcal N=2$ supersymmetry. There are eight real supercharges. We use an $\mathcal N=1$ superfield notation that makes four manifest; the remaining four are also symmetries of the specified action.

The gauge group is $SU(2)$, with rank one and three generators $t^a=\sigma_a/2$. Retain the ordinary fundamental trace $\operatorname{Tr}_2(t^at^b)=\delta^{ab}/2$. The physical fields are one gauge field $A_\mu$, two Weyl fermions $\lambda,\psi$, and one complex scalar $\phi$, all valued in this adjoint algebra. In the notation already introduced, they belong to a vector $V$ and an adjoint chiral $\Phi$.

Pure means that there are no additional hypermultiplets. It does not remove the scalar or either fermion from the vector multiplet. An ordinary nonsupersymmetric Yang–Mills theory with just a gauge field would be a different theory. This field organization is the one in Seiberg–Witten, (2.1).

Choose zero theta angle for the following classical calculation. With the canonical fields and field strength $\mathcal W_\alpha$ of the prerequisite, the action is

$$\begin{aligned} S={}&\int d^4x\,d^4\theta\; 2\operatorname{Tr}_2 \left(\Phi^\dagger e^{2gV}\Phi e^{-2gV}\right)\\ &+\left[ \frac14\int d^4x\,d^2\theta \sum_{a=1}^3\mathcal W^{a\alpha}\mathcal W^a_\alpha +\text{h.c.}\right], \qquad W_{\rm superpotential}=0 . \end{aligned} \tag{SW01.1}$$

Here $g>0$ is dimensionless; it appears in covariant derivatives and interactions, while the component kinetic terms are canonical. The formula fixes the fermion interactions as well as the bosonic terms. Its relative coefficients are those of the N=2 action, not independent adjustable couplings. Tachikawa, (2.1.10)–(2.1.11) gives the same construction in a different field normalization, translated in the prerequisite.

Eliminating the auxiliaries gives the bosonic density

$$\mathcal L_{\rm bos} =-\frac14 F_{\mu\nu}^aF^{a\mu\nu} +2\operatorname{Tr}_2\left[(\mathscr D_\mu\phi)^\dagger \mathscr D^\mu\phi\right] -g^2\operatorname{Tr}_2[\phi,\phi^\dagger]^2 , \tag{SW01.2}$$

where $\mathscr D_\mu\phi=\partial_\mu\phi-ig[A_\mu,\phi]$ and $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu-ig[A_\mu,A_\nu]$. The commutator $[\phi,\phi^\dagger]$ is Hermitian, so its squared trace is nonnegative. Thus the final term is minus a nonnegative potential. Each term has mass dimension four: $[\phi]=[A_\mu]=1$, $[F_{\mu\nu}]=2$ and $[g]=0$.

In Seiberg–Witten, (2.6), the displayed potential has an overall $1/g^2$. Its scalar is $\phi_{\rm SW}=g\phi$ in the present canonical normalization. Substitution gives precisely $g^2\operatorname{Tr}[\phi,\phi^\dagger]^2$. Comparing the prefactors without transforming the fields would give the wrong masses.

First calculation: find a charged threshold

Take the constant background

$$\phi_0=v\,t^3,\qquad v>0,\qquad A_\mu=0 .$$

Its commutator with its conjugate vanishes. The scalar auxiliary is zero because there is no superpotential, and the gauge auxiliary vanishes on this configuration. It is a classical supersymmetric vacuum. We are choosing one representative to calculate around; classifying all vacua and their gauge identifications is a separate problem.

The generator $t^3$ commutes with $\phi_0$, while $t^1,t^2$ do not. The associated gauge-field mass term comes directly from the scalar kinetic term. Using $[t^1,t^3]=-it^2$ and $[t^2,t^3]=it^1$,

$$\mathscr D_\mu\phi_0 =gv(A_\mu^2t^1-A_\mu^1t^2),$$

and hence

$$2\operatorname{Tr}_2\left[ (\mathscr D_\mu\phi_0)^\dagger\mathscr D^\mu\phi_0\right] =g^2v^2\left(A_\mu^1A^{1\mu}+A_\mu^2A^{2\mu}\right).$$

For a canonically normalized real vector, the mass term is $M^2A_\mu A^\mu/2$. Therefore

$$M_W^2=2g^2v^2,\qquad M_W=\sqrt2\,gv . \tag{SW01.3}$$

The complex fields $W_\mu^\pm=(A_\mu^1\mp iA_\mu^2)/\sqrt2$ carry opposite charges under the remaining $U(1)$. They are conjugates, not two independent complex vector fields. The neutral field $A_\mu^3$ remains massless. The mass has the correct dimension, and tends to zero as the background tends to zero.

The scalar fluctuations check the coefficient. Write

$$\phi=v t^3+\frac1{\sqrt2}\sum_{a=1}^3(x_a+iy_a)t^a .$$

The real variables have canonical kinetic terms $\frac12\sum_a[(\partial x_a)^2+(\partial y_a)^2]$. To first order, $[\phi,\phi^\dagger]=\sqrt2v(-y_2t^1+y_1t^2)$. The quadratic potential is consequently

$$U^{(2)} =g^2v^2(y_1^2+y_2^2) =\frac12M_W^2(y_1^2+y_2^2). \tag{SW01.4}$$

The $x_1,x_2$ directions are gauge-orbit directions eaten by the massive vectors. The two real fields $x_3,y_3$ remain massless. Thus the charged bosons have $2\times3+2=8$ physical modes; the neutral sector has a photon’s two modes and two scalar modes. The total is still twelve. This agrees with $4\dim SU(2)=12$ before the Higgs mechanism; it is not the count obtained by multiplying by the gauge rank.

This calculation is semiclassical. It is useful when the coupling at the charged threshold is small. The quantum theory has a dynamically generated strong scale $\Lambda$; far enough from that scale the classical threshold is the leading approximation. Nothing here extrapolates (SW01.3) through a strongly coupled region using an unchanged canonical scalar coordinate.

Specify what low energy means

Around this weakly coupled vacuum, experiments with characteristic momenta $E\ll M_W$ cannot create the charged vector multiplets on shell. Their virtual effects still change the interactions of the light fields. A local description that removes them keeps the neutral N=2 vector multiplet: a photon, a complex scalar, and two Weyl fermions.

“Local” also restricts the scalar excursions. Expanding about this vacuum while allowing the scalar to wander into a region with a new light charged state would invalidate the chosen field content. An energy cutoff that works at one vacuum need not work everywhere in the family.

The action can be organized by derivatives of the retained fields. A two-derivative bosonic term includes a scalar kinetic term or $F_{\mu\nu}F^{\mu\nu}$; each field strength counts as one derivative. The adjective does not mean quadratic in the fields. A coefficient depending on the scalar generates interactions when expanded around a background.

For example, an additional local photon interaction of the form

$$\Delta\mathcal L =\frac{c}{M_*^4} \left(F_{\mu\nu}F^{\mu\nu}\right)^2 \tag{SW01.5}$$

has four derivatives and dimension four, with dimensionless $c$. Here $M_*$ denotes a heavy scale appropriate to that local expansion, and the photon has been normalized canonically at the chosen vacuum. This displayed bosonic term alone is not a proposed supersymmetric action; its required completion and coefficient would need a further calculation. Its local four-photon contribution scales as $c(E/M_*)^4$.

Knowing a two-derivative action exactly does not determine that extra coefficient. It also does not remove the derivative expansion’s error. A controlled use specifies both the accuracy of the retained coefficients and which orders in momentum have been discarded. These are separate questions.

What the exact solution supplies

N=2 supersymmetry relates the light scalar and gauge couplings. On a suitable local patch they can be packaged in a holomorphic function called the prepotential, $\mathcal F$. It is distinct from the superpotential $W$ in (SW01.1) and from a free energy.

In a local scalar coordinate $a$ adapted to that supersymmetry, the relations include

$$a_D=\frac{d\mathcal F}{da},\qquad \tau_{\rm eff}=\frac{d^2\mathcal F}{da^2} =\frac{da_D}{da},\qquad \tau_{\rm eff}=\frac{\theta_{\rm eff}}{2\pi} +\frac{4\pi i}{g_{\rm eff}^2}. \tag{SW01.6}$$

Here $a_D$ is a derived complex function, and $\tau_{\rm eff}$ specifies the local photon coupling and theta angle. A physical electric description requires $\operatorname{Im}\tau_{\rm eff}>0$. The detailed action and the special choice of coordinate will be derived when we study the low-energy theory. These relations are the target data, as in Seiberg–Witten, (2.7)–(2.10).

The solution determines these functions with quantum effects included, using compatible local electric and magnetic descriptions. It also supplies central-charge data entering supersymmetric particle mass bounds. A bound tells us the mass of a state if that state exists and saturates it; it does not by itself enumerate all stable particles. The original paper treats this distinction in its section 4 discussion after (4.5).

Near a point where a charged particle becomes light, a description containing only the neutral multiplet is insufficient. One must retain the additional light fields and use appropriate variables. A singular coefficient can therefore indicate missing light physics, rather than permission to apply the same truncated action at all scales.

Our starting question is now precise: how do the leading light-field couplings vary across the supersymmetric vacua, and which descriptions remain valid in each region? Answering it does not require an arbitrary-energy S-matrix, every higher-derivative coefficient, or every unprotected correlation function. Nor does pure N=2 theory automatically acquire the confining vacua of a separately deformed N=1 theory.

Exercises

1. Change normalization without changing a mass

Define $\varphi=g\phi$ while retaining the canonical gauge field $A_\mu$ and the same fixed coupling $g$. Rewrite the scalar kinetic term and potential. At $\varphi_0=b t^3$, find the charged-vector mass. Explain the error in reading $b$ as the previous canonical expectation value $v$.

Hint 1

Both $\mathscr D\phi$ and $[\phi,\phi^\dagger]$ acquire powers of $g^{-1}$. The gauge covariant derivative still contains $gA_\mu$.

Hint 2

Keep the factor $1/g^2$ in the scalar kinetic term while evaluating its term quadratic in $A_\mu$.

Solution

Substitution gives

$$\mathcal L_{\rm scalar} =\frac2{g^2}\operatorname{Tr}|\mathscr D\varphi|^2 -\frac1{g^2}\operatorname{Tr}[\varphi,\varphi^\dagger]^2.$$

At the background, the factor $g^2b^2$ from the covariant derivative cancels the kinetic prefactor. The vector mass term is $b^2[(A^1)^2+(A^2)^2]$, so $M_W=\sqrt2|b|$. Since $b=gv$, this agrees with (SW01.3).

Using the formula $\sqrt2 g|b|$ would apply a canonical-field result to a noncanonical scalar. Conversely, discarding the kinetic prefactor would change the theory. An overall field normalization is not an independently measurable mass shift.

2. Does giving the scalar a mass keep the same pure theory?

Add $W_{\rm def}=\mu\operatorname{Tr}_2\Phi^2$ with $\mu\ne0$. Derive its additional scalar potential in canonical components. What happens to the classical constant-scalar supersymmetric vacua? Does this preserve the original eight supercharges?

Hint 1

The trace makes $W_{\rm def}=(\mu/2)\sum_a(\Phi^a)^2$. Use the canonical chiral auxiliary equations.

Hint 2

Test the F equation as well as the commutator potential. Distinguish a symmetry of the new action from supersymmetry preserved by its vacuum.

Solution

For each adjoint component, $W_a=\mu\phi^a$ and $F^a=-\overline\mu\,\overline{\phi^a}$. The full classical potential becomes

$$U=g^2\operatorname{Tr}[\phi,\phi^\dagger]^2 +|\mu|^2\sum_a|\phi^a|^2 .$$

Both terms are nonnegative. Vanishing of the F terms requires $\phi=0$, which also makes the gauge auxiliary vanish. Thus the classical scalar vacuum family is lifted to the origin. The adjoint chiral scalar and its Weyl fermion have mass $|\mu|$ there; the gauge multiplet has not acquired this same mass.

This superpotential preserves the manifest N=1 supersymmetry but breaks the second supersymmetry of the action. The classical vacuum preserves the remaining four real supercharges. It is not spontaneous breaking of those four, and it is not the original pure N=2 theory with an irrelevant field removed. Quantum infrared dynamics and possible confinement require additional analysis; the classical equation at the origin cannot establish them.

3. Exact coefficients and a shrinking domain

Suppose a two-derivative coupling has been found exactly, and a separate calculation gives a nonzero four-photon contact coefficient $c/M_*^4$ as in (SW01.5). Compare its contribution at $E=M_*/10$ and $E=M_*/2$. Then hold $E$ fixed while changing the vacuum so that the lightest omitted mass decreases from $100E$ to $E/10$. Can exact knowledge of the two-derivative coefficient justify the same reduced description throughout?

Hint 1

Count the four powers of external momentum in the local contact term. This is an absolute scaling estimate for that contribution, not a ratio to an unspecified full amplitude.

Hint 2

The energy hierarchy concerns the lightest degree of freedom actually removed from the action, rather than the accuracy with which a retained coefficient is known.

Solution

Up to the same angular and polarization factor, the two contact contributions scale as $c\,10^{-4}$ and $c/16$; the latter is $625$ times larger. A small parameter at the first energy does not imply a small correction at the second. No fractional error for a full scattering amplitude follows without its other terms and possible cancellations.

Initially $E/M_{\rm omitted}=1/100$, a useful hierarchy. Finally it is $10$; the omitted state can be produced and cannot be represented solely through a local expansion in small external momenta. It must be included explicitly, possibly in a different electric or magnetic description. Exact two-derivative couplings do not restore a failed scale separation. Even the first hierarchy must be supplemented by control of field excursions and of the relevant expansion coefficients.

You can now specify the pure theory, derive its elementary charged threshold, and explain the scope of its exact low-energy solution. The next physical task is to classify the scalar vacua and their gauge identifications. That turns the chosen background used here into the space on which the quantum coupling functions live.

Continue with Find the classical Coulomb branch of SU(2) to solve the full vacuum equation, remove gauge copies and identify the invariant coordinate with this normalization.