N=2 Multiplets, Lagrangians, and Vacuum Branches

Four-dimensional $\mathcal N=2$ gauge theories are built from vector multiplets and hypermultiplets. In $\mathcal N=1$ language, the extended supersymmetry fixes the adjoint–matter superpotential and organizes three moment maps. These constraints separate the vacuum space into Coulomb, Higgs, and mixed branches with different low-energy fields and different protection properties.

Required background. Extended supersymmetry and R-symmetry fixes the algebra, while supersymmetric Yang–Mills fixes the $\mathcal N=1$ gauge conventions. Helpful background. F- and D-flat quotients supplies the vacuum construction.

N=2 multiplets in N=1 superspace

An $\mathcal N=2$ vector multiplet for gauge group $G$ decomposes as

$$\mathcal V_{\mathcal N=2} \longrightarrow V\oplus\Phi,$$

where $V$ is an $\mathcal N=1$ vector multiplet and $\Phi$ is an adjoint chiral multiplet. On shell it contains a gauge field, one complex adjoint scalar $\phi$, and two Weyl gauginos.

A hypermultiplet in a complex representation $R$ decomposes as

$$\mathcal H \longrightarrow Q\oplus\widetilde Q,$$

with $Q\in R$ and $\widetilde Q\in\overline R$. The two complex scalars form a doublet of $SU(2)_R$ after one is conjugated appropriately.

For a simple gauge group and canonical hypermultiplet normalization, the superspace action has the schematic form

$$\begin{aligned} S={}&\operatorname{Im}\!\left[ \frac{\tau}{8\pi}\int d^4x\,d^2\theta\, \operatorname{Tr}W^\alpha W_\alpha \right]\\ &+\int d^4x\,d^4\theta\left[ Q^\dagger e^{2V}Q +\widetilde Qe^{-2V}\widetilde Q^\dagger +\frac{1}{g^2}\operatorname{Tr} \Phi^\dagger e^{2V}\Phi e^{-2V} \right]\\ &+\int d^4x\,d^2\theta\left[ \sqrt2\,\widetilde Q\Phi Q +\widetilde QmQ \right]+\text{h.c.} \end{aligned}$$

The placement of $g$ between $V$, $\Phi$, and the generators varies by convention; the relative Yukawa coupling is fixed by the second supersymmetry. The mass $m$ is the scalar expectation value of a background flavor vector multiplet. An FI parameter is an $SU(2)_R$ triplet and is available only for abelian gauge factors.

The scalar potential and three moment maps

For vanishing masses, eliminating auxiliary fields gives a nonnegative potential whose zeros can be organized as

$$\begin{aligned} (\phi Q)&=0, & (\widetilde Q\phi)&=0,\\ \mu_{\mathbb C}^a&=\widetilde Q T^aQ=\zeta_{\mathbb C}^a, & \mu_{\mathbb R}^a&= Q^\dagger T^aQ-widetilde QT^a\widetilde Q^\dagger +[\phi,\phi^\dagger]^a=\zeta_{\mathbb R}^a. \end{aligned}$$

For nonzero flavor mass, replace $\phi Q=0$ by $(\sqrt2\phi+m)Q=0$ and similarly for $\widetilde Q$. Generator, $\sqrt2$, and FI signs follow the displayed superpotential convention.

When $\phi=0$, the three real equations $(\mu_{\mathbb R},\operatorname{Re}\mu_{\mathbb C},\operatorname{Im}\mu_{\mathbb C})=\vec\zeta$ are the hyperkähler moment-map equations. Quotienting by $G$ produces a hyperkähler Higgs branch.

Coulomb branches

Set hypermultiplet expectation values to zero. The vacuum condition is

$$[\phi,\phi^\dagger]=0,$$

so $\phi$ can be conjugated into a Cartan subalgebra. Dividing by the Weyl group gives classically

$$\mathcal B_{\mathrm C}^{\mathrm{cl}} =\mathfrak t_{\mathbb C}/W.$$

Gauge-invariant coordinates can be chosen as Casimirs

$$u_k=\langle\operatorname{Tr}\phi^k\rangle.$$

At a generic point, $G$ is broken to $U(1)^r$, where $r=\operatorname{rank}G$. Massive $W$ bosons and charged matter are integrated out, leaving $r$ abelian vector multiplets. The complex dimension of the Coulomb branch is $r$.

Quantum effects generally correct its metric and shift its singular loci, but $\mathcal N=2$ supersymmetry retains rigid special Kähler structure. In an asymptotically free theory the anomalous $U(1)_r$ leaves a discrete subgroup and introduces the dynamical scale $\Lambda$. The pure-gauge moduli and abelian low-energy action are set up in Seiberg and Witten 1994, §§2.2–2.3.

Higgs branches

On a pure Higgs branch, hypermultiplets acquire expectation values and the relevant gauge group is Higgsed. At zero masses and for a regular free quotient,

$$\mathcal M_{\mathrm H} =\mathbb H^{n_H}/\!/\!/G,$$

with

$$\dim_{\mathbb R}\mathcal M_{\mathrm H} =4(n_H-\dim G).$$

Here $n_H$ is the quaternionic dimension of the hypermultiplet representation. Stabilizers, dependent moment maps, or disconnected components require a stratum-by-stratum count.

The Higgs-branch metric is hyperkähler and, in a rigid four-dimensional $\mathcal N=2$ theory under the standard assumptions, does not receive quantum corrections. Its complex presentation can still vary with the chosen complex structure, and global quotient data remain important. The supersymmetric sigma-model origin of the target-space restrictions is explained in Alvarez-Gaumé and Freedman 1981, pp. 443–451, and the hyperkähler quotient construction is developed in Hitchin, Karlhede, Lindström, and Roček 1987, pp. 535–589.

For $U(1)$ with $n$ charge-one hypermultiplets, the triplet quotient at nonzero FI parameter gives

$$T^*\mathbb{CP}^{n-1},$$

of real dimension $4(n-1)$. Sending the FI triplet to zero collapses the exceptional cycle and restores a singular cone.

Mixed branches

A mixed branch leaves an unbroken subgroup $H\subset G$ whose vector multiplet scalars vary on a Coulomb factor while hypermultiplets Higgs the complementary generators. Locally near a generic point, the branch can resemble

$$\mathcal B_{\mathrm C}(H)\times\mathcal M_{\mathrm H}^{\mathrm{transverse}},$$

possibly divided by a discrete group. At branch intersections additional vector or hypermultiplets become massless, so a sigma model using only one factor is incomplete.

Mass parameters can lift Higgs directions or move them to loci where eigenvalues of $\phi$ cancel flavor masses. FI parameters can resolve Higgs singularities and lift portions of a Coulomb branch for an abelian factor. Each statement must name the parameter chamber.

R-symmetry and branch operators

The classical R-symmetry is

$$SU(2)_R\times U(1)_r.$$

The Coulomb scalar $\phi$ is an $SU(2)_R$ singlet with nonzero $U(1)_r$ charge, so Coulomb-branch chiral operators are $SU(2)_R$ singlets. Higgs-branch coordinates sit in $SU(2)_R$ multiplets and are neutral under the appropriate $U(1)_r$ assignment. This distinction underlies their different shortening conditions and metric protection.

Masses transform as background vector-multiplet scalars and are complex. FI parameters transform as an $SU(2)_R$ triplet. Treating them as interchangeable real deformations breaks the extended supersymmetry bookkeeping.

Which data are protected

Branch	Generic light multiplets	Geometry	Quantum status
Coulomb	Abelian vector multiplets	Rigid special Kähler	Metric receives one-loop and instanton corrections; encoded by exact low-energy data.
Higgs	Hypermultiplets	Hyperkähler	Intrinsic two-derivative metric protected in the standard rigid setting.
Mixed	Both	Stratified combination	Each factor follows its own protection; intersections require all light fields.

Protection of a metric does not prove that a branch exists for every mass or FI value, and it does not eliminate higher-derivative corrections.

Common pitfalls

Choosing an arbitrary Yukawa coupling. The coefficient relating $\widetilde Q\Phi Q$ to the gauge coupling is fixed after field normalization by $\mathcal N=2$ supersymmetry.

Counting a Higgs quotient as free when stabilizers remain. The simple $4\dim G$ subtraction assumes a regular level and free action.

Using Coulomb-branch corrections on the Higgs metric. The two branches sit in different multiplets and obey different nonrenormalization statements.

Exercises

For $SU(2)$ with two fundamental hypermultiplets, use quaternionic dimension counting to find the real dimension of the fully Higgsed branch at a regular point.

Solution

Each $SU(2)$ fundamental hypermultiplet has quaternionic dimension two, so two flavors give $n_H=4$. Since $\dim SU(2)=3$,

$$\dim_{\mathbb R}\mathcal M_H=4(4-3)=4.$$

This assumes the gauge action is fully Higgsed and the moment-map level is regular. At the cone tip the stabilizer grows and the smooth-stratum count does not describe the tangent space.

References

• Alvarez-Gaumé, Luis, and Daniel Z. Freedman. “Geometrical Structure and Ultraviolet Finiteness in the Supersymmetric Sigma Model.” Communications in Mathematical Physics 80 (1981): 443–451. doi:10.1007/BF01208280.
• Hitchin, Nigel J., Anders Karlhede, Ulf Lindström, and Martin Roček. “Hyperkähler Metrics and Supersymmetry.” Communications in Mathematical Physics 108 (1987): 535–589. doi:10.1007/BF01214418.
• 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. arXiv:hep-th/9407087.