Three-Dimensional Supersymmetry Algebras and Multiplets

Three-dimensional supersymmetry starts with a two-real-component Lorentzian spinor, but the multiplets most useful for gauge dynamics are best organized by four or eight real supercharges. The vector multiplet then contains a physical real scalar, background vector multiplets generate real masses, and the conserved flux current supplies a topological symmetry. These features are the algebraic source of the Coulomb branches and deformation maps used throughout the chapter.

Required background. We use the classification of spinors and reality conditions and the distinction among off-shell, on-shell, and modulo-gauge closure. Helpful background. The later potential formulas use gauge–matter couplings and F- and D-terms.

Spinors and the three-dimensional super-Poincaré algebra

Work in oriented Lorentzian signature $(+--)$, with $\epsilon^{012}=+1$. The connected spin group is $\operatorname{Spin}(1,2)\simeq SL(2,\mathbb R)$, whose fundamental is a two-real-dimensional Majorana representation. Let $C_{\alpha\beta}=-C_{\beta\alpha}$ lower spinor indices and choose $(\gamma^\mu C)_{\alpha\beta}$ symmetric. The minimal algebra is

$$\{Q_\alpha,Q_\beta\}=2(\gamma^\mu C)_{\alpha\beta}P_\mu .$$

There are two real supercharges. For $\mathcal N$ Majorana charges, the general scalar central extension has the form

$$\{Q^I_\alpha,Q^J_\beta\} =2\delta^{IJ}(\gamma^\mu C)_{\alpha\beta}P_\mu +2C_{\alpha\beta}Z^{IJ}, \qquad Z^{IJ}=-Z^{JI}.$$

The antisymmetries of $C$ and $Z$ make the last term symmetric under interchange of the two pairs $(I,\alpha)$ and $(J,\beta)$. Positivity in a rest frame gives BPS bounds involving the skew eigenvalues of $Z^{IJ}$. Extended algebras can also carry tensorial charges associated with extended objects; which appear depends on the allowed defects and boundary conditions.

For $\mathcal N=2$, combine $Q^1,Q^2$ into a complex charge $Q$ of $U(1)_R$ charge $+1$. The automorphism rotating the two real charges is $SO(2)_R\simeq U(1)_R$. There are four real supercharges. For $\mathcal N=4$, there are eight, and the R-symmetry Lie algebra is

$$\mathfrak{so}(4)_R\simeq \mathfrak{su}(2)_H\oplus\mathfrak{su}(2)_C.$$

The faithfully acting global group is theory dependent: centers can be combined with fermion parity or a gauge-group center, so the quotient cannot be inferred from the superalgebra alone. The labels $H$ and $C$ anticipate the action on Higgs- and Coulomb-branch coordinates. They are not interchangeable inside a fixed ultraviolet Lagrangian; mirror symmetry will exchange them in the infrared Intriligator and Seiberg 1996, §§1–2.

The N=2 central charge and its BPS bound

Set $Z^{12}=Z$ and define $Q=(Q^1-iQ^2)/\sqrt2$. The real algebra above becomes

$$\{Q_\alpha,Q_\beta\} =\{\bar Q_\alpha,\bar Q_\beta\}=0, \qquad \{Q_\alpha,\bar Q_\beta\} =2(\gamma^\mu C)_{\alpha\beta}P_\mu +2iC_{\alpha\beta}Z,$$

where reversing the definition of the complex supercharge reverses the sign assigned to $Z$. In a massive rest frame, a unitary change of spinor basis turns the positive mixed anticommutators into $2(M+Z)$ and $2(M-Z)$. Positivity therefore gives

$$M\geq |Z|.$$

This is more than a formal bound. Dimensional reduction of four-dimensional $\mathcal N=1$ identifies the discarded momentum $P_3$ with the three-dimensional central charge, while the fourth gauge-field component becomes $\sigma$. A constant scalar in a background vector multiplet is the same deformation for a flavor symmetry: a state of flavor charge $q$ receives $Z=qm$. A free chiral with only this real mass has boson and fermion mass $|m|$ and saturates the bound. If a compatible complex superpotential mass $\mu$ is also present, the physical mass becomes $\sqrt{m^2+|\mu|^2}$ and the state is no longer shortened by $Z=m$ alone Aharony et al. 1997, §2.3, pp. 6–8.

The N=2 chiral and vector multiplets

An off-shell chiral multiplet in a complex representation $R$ is

$$\Phi=(\phi,\psi_\alpha,F),$$

with a complex scalar, a complex two-component spinor, and a complex auxiliary scalar. Its off-shell bosonic and fermionic component counts are both four real components. Chirality is naturally inherited from four-dimensional $\mathcal N=1$ superspace after suppressing one coordinate; in three dimensions it is a superspace constraint, not a statement about Lorentz chirality.

An off-shell vector multiplet is

$$V=(A_\mu,\sigma,\lambda_\alpha,\bar\lambda_\alpha,D),$$

where $A_\mu$, the real scalar $\sigma$, and the real auxiliary $D$ are Lie-algebra valued, while $\lambda$ is a complex two-component gaugino. Before equations of motion, $A_\mu$ contributes $3-1=2$ real components modulo infinitesimal gauge transformations; together with $\sigma$ and $D$, this matches the four real components of $\lambda$.

For a chiral whose scalar has $R$-charge $r$, the charges of $(\phi,\psi,F)$ are $(r,r-1,r-2)$. The vector bosons $A_\mu$ and $\sigma$ and the auxiliary $D$ are $R$-neutral; $\lambda$ and $\bar\lambda$ have opposite unit charges. In the connection normalization $D_\mu=\partial_\mu-i\rho(A_\mu)$ used in the next page, the engineering dimensions are

field or parameter	mass dimension
$\phi,\psi,F$	$1/2,1,3/2$
$A_\mu,\sigma,\lambda,D$	$1,1,3/2,2$
$e^2,m,\zeta$	$1,1,1$

The dimensions of $A_\mu$ and $\sigma$ would instead be $1/2$ after a canonical rescaling that moves $e$ into the covariant derivative. Mixing those two normalizations is a common source of apparently inconsistent mass dimensions.

In Wess–Zumino gauge a supersymmetry transformation does not remain in Wess–Zumino gauge. Its compensating gauge transformation is essential. Thus, on any covariant field $X$,

$$[\delta_{\varepsilon_1},\delta_{\varepsilon_2}]X =v^\mu D_\mu X+\delta_{\rm gauge}(\Lambda)X +\delta_R(\rho)X+\delta_F(\omega)X,$$

with $v^\mu$ a spinor bilinear. In flat space without background sources the last two terms vanish. The auxiliary fields $F$ and $D$ make this closure off shell; eliminating them turns the same statement into on-shell closure. On $A_\mu$ itself the translation and gauge pieces can be rearranged into $v^\nu F_{\nu\mu}$, which is why “closure modulo gauge” is not optional bookkeeping.

These multiplets, their real central charges, and their use as three-dimensional gauge-theory coordinates are developed systematically in Aharony et al. 1997, §2, pp. 3–10.

The Yang–Mills vector has one propagating gauge polarization in three dimensions, and $\sigma$ supplies the second on-shell bosonic degree of freedom. A pure Chern–Simons vector is different: the gauge field and the remaining vector-multiplet fields are topological or auxiliary in the absence of a Yang–Mills regulator.

Real masses, FI parameters, and the topological current

Let $G_F$ be an ordinary flavor symmetry and weakly gauge it by a nondynamical vector multiplet. Giving its scalar a constant value in the Cartan,

$$\sigma_F=m,$$

produces a real mass. A component of weight $w$ experiences the effective mass $w(m)$; for a gauge weight $\rho$ as well, the mass is $\rho(\sigma)+w(m)$. Unlike a complex mass in the superpotential, a real mass preserves the phase rotation generated by the corresponding flavor Cartan and affects parity-odd contact terms when its sign changes.

Every Abelian gauge factor has a classically conserved topological current

$$j_J^\mu=\frac{1}{2\pi}\epsilon^{\mu\nu\rho}\partial_\nu a_\rho =\frac{1}{4\pi}\epsilon^{\mu\nu\rho}F_{\nu\rho}, \qquad \partial_\mu j_J^\mu=0.$$

The normalization makes a unit-flux monopole carry $U(1)_J$ charge one. For a non-Abelian $U(N)$ factor, replace $F$ by $\operatorname{Tr}F$ in the fundamental trace convention. For a simple simply connected group there is generally no continuous current of this form; global form may instead leave a discrete topological symmetry.

Let $m_J$ be the scalar in the background vector multiplet for $U(1)_J$. With the BF convention $\frac{1}{2\pi}D\,m_J$ and the FI convention $-\zeta D$ used in this chapter,

$$m_J=-2\pi\zeta.$$

Both ordinary real masses and $\zeta$ have mass dimension one. A BPS particle or vortex can then contribute

$$Z=q_Fm+q_Jm_J+Z_{\rm complex} =q_Fm-2\pi q_J\zeta+Z_{\rm complex},$$

where the sign follows from the chosen BF and FI conventions. In particular, a vortex of flux $q_J=n$ has $|Z|=2\pi|\zeta n|$. This formula is a diagnostic: a duality exchanging ordinary and topological currents must exchange an ordinary mass with the normalized topological mass $m_J$, or equivalently with the FI parameter including this fixed sign and factor.

Some references normalize the topological current without $1/(2\pi)$ and consequently write the FI contribution to $Z$ without the factor $2\pi$. The invariant comparison is the product “charge times background scalar”: after rescaling the current, its charge and source rescale inversely. In the normalization on this page, unit magnetic flux means $q_J=1$ and fixes the coefficient above.

Worked example: a charge-one chiral

Take $U(1)$ with one chiral of gauge charge $+1$, no superpotential, background real mass $m$, and vector scalar $\sigma$. Its scalar and fermion acquire mass from the same combination

$$M_{\rm eff}=\sigma+m.$$

The scalar potential contains $M_{\rm eff}^2|\phi|^2$. A Higgs vacuum with $\phi\neq0$ therefore requires $\sigma=-m$, in addition to the D-term equation. By contrast, on a Coulomb region with $\phi=0$, changing the sign of $M_{\rm eff}$ changes the parity-odd effective action of the fermion. The locus $\sigma=-m$ is consequently both a possible Higgs attachment point and a wall between contact-term chambers.

N=4 decompositions and protected branches

In $\mathcal N=2$ language,

$$\mathcal V_{\mathcal N=4}=V\oplus\Phi_{\rm adj}, \qquad \mathcal H_{\mathcal N=4}=Q\oplus\widetilde Q,$$

where the two chirals in a hypermultiplet transform in conjugate gauge representations. The superpotential coupling $\widetilde Q\Phi_{\rm adj}Q$ and its Yukawa partners are fixed by $\mathcal N=4$. The three real scalars in the vector multiplet form a triplet of $SU(2)_C$, while the four real hypermultiplet scalars form a doublet of $SU(2)_H$. Correspondingly, Coulomb-branch operators transform under $SU(2)_C$ and Higgs-branch operators under $SU(2)_H$.

Real masses and FI parameters each enlarge to triplets. Masses are background-vector scalars and transform as a triplet of $SU(2)_C$; FI parameters form a triplet of $SU(2)_H$. Choosing only one component displays an $\mathcal N=2$ subalgebra and can obscure this covariance. Generic $\mathcal N=2$ superpotentials, independent Chern–Simons levels, or arbitrary R-charge assignments do not preserve $\mathcal N=4$.

The phrase “classical Coulomb branch” is also potentially misleading. In an $\mathcal N=4$ gauge description the Coulomb-branch metric and operator relations receive quantum corrections, whereas the Higgs branch is described by a hyperkähler quotient and is protected. A mirror description reverses which geometry is manifestly classical.

Euclidean continuation

After Wick rotation to three-dimensional Euclidean space, $\operatorname{Spin}(3)=SU(2)$ and a two-component spinor is pseudoreal. A Lorentzian Majorana condition does not continue as an ordinary reality condition on a single Euclidean spinor. In Euclidean functional integrals one therefore treats barred and unbarred spinors as independent complex variables and chooses an integration contour that returns the desired Lorentzian theory.

The same warning applies to auxiliaries in localization: a convenient Euclidean contour may take $D$ along an imaginary direction even though Lorentzian $D$ is real. Algebraic supersymmetry variations can be complexified, but Hermiticity, positivity, and BPS bounds must be interpreted after a reality contour is specified.

Common pitfalls

Confusing $\mathcal N$ across dimensions. Three-dimensional $\mathcal N=2$ has four real supercharges and is the direct reduction of four-dimensional $\mathcal N=1$. Three-dimensional $\mathcal N=4$, not $\mathcal N=2$, has eight real supercharges.

Treating $U(1)_J$ as automatic. Its existence and normalization depend on the global gauge group, allowed bundles, and monopole effects. A monopole superpotential can explicitly break a continuous topological symmetry.

Using Euclidean “Majorana spinors” without a contour. Euclidean manipulations normally complexify the spinors. A Lorentzian reality statement cannot simply be copied into the Euclidean path integral.

Exercises

1. Count the off-shell real degrees of freedom in an $\mathcal N=2$ chiral multiplet and in a vector multiplet modulo gauge transformations.

Solution

For a chiral, $\phi$ and $F$ contribute two real components each, totaling four; the complex two-component spinor also has four. For a vector, $A_\mu$ has three components minus one gauge parameter, while $\sigma$ and $D$ contribute one each, again totaling four; the complex two-component gaugino has four. Equations of motion remove the auxiliaries and halve the propagating counts in the usual way.

2. A $U(1)$ theory has two chirals of gauge charges $+1$ and $-1$, with background flavor weights $+1$ and $+1$. Write their effective real masses at vector scalar $\sigma$ and flavor mass $m$. Where can each become massless?

Solution

The masses are $M_+=\sigma+m$ and $M_-=-\sigma+m$. The first field is massless at $\sigma=-m$ and the second at $\sigma=m$. These two walls divide the Coulomb direction into chambers with potentially different induced Chern–Simons contact terms.

3. For an $\mathcal N=2$ state with ordinary flavor charge $q_F$ and topological charge $q_J$, use the rest-frame algebra to find its BPS lower bound in the presence of a real mass $m$ and FI parameter $\zeta$. Assume there is no additional central contribution.

Solution

The two positive rest-frame anticommutators have eigenvalues proportional to $M+Z$ and $M-Z$, so both are nonnegative only when $M\geq|Z|$. In the conventions of this page,

$$Z=q_Fm-2\pi q_J\zeta,$$

and hence

$$M\geq\left|q_Fm-2\pi q_J\zeta\right|.$$

Saturation removes one complex supercharge from the long massive representation. A pure unit-flux vortex has $q_F=0$, $q_J=1$, and saturated mass $2\pi|\zeta|$, agreeing with the Bogomolny calculation on the action page.

References

• Aharony, O., Hanany, A., Intriligator, K., Seiberg, N., and Strassler, M. J. (1997), “Aspects of $N=2$ Supersymmetric Gauge Theories in Three Dimensions,” Nuclear Physics B 499, 67–99. doi:10.1016/S0550-3213(97)00323-4. Open PDF
• Intriligator, K., and Seiberg, N. (1996), “Mirror Symmetry in Three Dimensional Gauge Theories,” Physics Letters B 387, 513–519. doi:10.1016/0370-2693(96)01088-X. Open PDF

Next steps

Use these multiplets to assemble Yang–Mills, Chern–Simons, BF, FI, and matter actions. Their fermions then force the parity-anomaly and contact-term constraints.