Two-Dimensional Supersymmetry Algebras, Multiplets, and Superspace

Two-dimensional supersymmetry is organized by chirality before it is organized by multiplet size. Independent right- and left-moving supercharges give $(p,q)$ algebras, two inequivalent R symmetries in the $(2,2)$ case, and two different notions of chirality for superfields. This page fixes the algebra and conventions used throughout the chapter and shows how the standard chiral, twisted-chiral, vector, and bounded $(0,2)$ multiplets follow from superspace constraints.

Required background. We use spinor reality conditions in different dimensions and the logic of superspace covariant derivatives and chirality constraints. Helpful background. Complex coordinates in two-dimensional CFT clarify the left–right terminology.

The $(p,q)$ algebra

Let $Q_+^I$, $I=1,\ldots,p$, have Lorentz spin $+\tfrac12$, and let $Q_-^A$, $A=1,\ldots,q$, have spin $-\tfrac12$. With $P_\pm=P_0\pm P_1$, the algebra without central extensions is

$$\{Q_+^I,(Q_+^J)^\dagger\}=2\delta^{IJ}P_+, \qquad \{Q_-^A,(Q_-^B)^\dagger\}=2\delta^{AB}P_-,$$

and mixed anticommutators vanish. Positivity gives $P_\pm\ge0$, hence $P_0\ge |P_1|$. A state annihilated by all right-moving supercharges has $P_+=0$; with our conventions it is purely left-moving. This statement depends on the displayed definition of $P_\pm$, not on the often inconsistent words “left” and “right.”

For $(2,2)$ it is useful to name complex supercharges $Q_\pm,\bar Q_\pm=Q_\pm^\dagger$. The most general translation-invariant central extension compatible with Lorentz spin can be written

$$\begin{aligned} \{Q_+,\bar Q_+\}&=2P_+,& \{Q_-,\bar Q_-\}&=2P_-,\\ \{Q_+,Q_-\}&=2\bar Z,& \{\bar Q_+,\bar Q_-\}&=2Z,\\ \{Q_+,\bar Q_-\}&=2\bar{\widetilde Z},& \{\bar Q_+,Q_-\}&=2\widetilde Z. \end{aligned}$$

$Z$ and $\widetilde Z$ are Lorentz scalars. In a Landau–Ginzburg theory a kink can carry $Z$ proportional to a difference of superpotential values. Twisted masses and twisted superpotentials naturally contribute to $\widetilde Z$. These charges are surface terms: they can be nonzero in soliton sectors even when they vanish on local fields in the trivial sector.

In the rest frame, an appropriate phase rotation of the four real supercharges diagonalizes their anticommutator matrix and yields the BPS inequality

$$M\ge |Z|$$

when only $Z$ is present. Saturation makes one complex linear combination null and shortens the representation. The exact numerical coefficient relating $Z$ to $\Delta W$ is action-normalization dependent; later pages use $Z=2\Delta W$ and state it explicitly.

Vector and axial R symmetries

Classically the $(2,2)$ algebra admits $U(1)_V\times U(1)_A$. Our charge convention is

Supercharge	$Q_+$	$Q_-$	$\bar Q_+$	$\bar Q_-$
$U(1)_V$	$+1$	$+1$	$-1$	$-1$
$U(1)_A$	$+1$	$-1$	$-1$	$+1$

The vector symmetry rotates both chiralities alike; the axial symmetry rotates them oppositely. A superpotential constrains matter R charges because the chiral measure $d\theta^+d\theta^-$ itself carries R charge. Gauge and sigma-model fermion measures can make an apparently classical R symmetry anomalous. Therefore an R-charge table is only the first half of a twisting argument; anomaly freedom is the second.

Dimensional reduction from four-dimensional $\mathcal N=1$ produces a $(2,2)$ algebra, but it hides a useful fact. The four-dimensional chiral multiplet reduces to a two-dimensional chiral multiplet, whereas the two components of the gauge field along reduced directions combine with the 2d vector multiplet scalar. Twisted-chiral multiplets are intrinsic to two-dimensional superspace and are not simply another name for reduced four-dimensional chirals.

$(2,2)$ superspace from the derivative algebra

Use coordinates $(x^\pm,\theta^\pm,\bar\theta^\pm)$ and define

$$D_\pm=\frac{\partial}{\partial\theta^\pm}-i\bar\theta^\pm\partial_\pm, \qquad \bar D_\pm=-\frac{\partial}{\partial\bar\theta^\pm}+i\theta^\pm\partial_\pm.$$

Then

$$\{D_\pm,\bar D_\pm\}=2i\partial_\pm, \qquad D_\pm^2=\bar D_\pm^2=0,$$

and all cross-chirality anticommutators vanish. Supercharges have the opposite sign in the terms proportional to $\partial_\pm$, so they anticommute with every $D$.

A chiral superfield obeys

$$\bar D_+\Phi=\bar D_-\Phi=0.$$

The compatibility condition is automatic because $\{\bar D_+,\bar D_-\}=0$. In chiral coordinates $y^\pm=x^\pm-i\theta^\pm\bar\theta^\pm$ its expansion begins

$$\Phi=\phi+\sqrt2\theta^+\psi_++\sqrt2\theta^-\psi_-+2\theta^+\theta^-F+\cdots,$$

where the omitted terms are fixed spacetime derivatives. Off shell it contains a complex scalar, two Weyl fermions, and a complex auxiliary field.

A twisted-chiral field instead satisfies the crossed constraints introduced in Gates, Hull, and Roček 1984:

$$\bar D_+Y=0, \qquad D_-Y=0.$$

It depends on $\theta^+$ and $\bar\theta^-$ rather than on $\theta^+$ and $\theta^-$. This crossed constraint is why ordinary and twisted superpotentials are integrated over different half-superspaces:

$$S_W=\int d^2x\,d\theta^+d\theta^-\,W(\Phi)+\text{c.c.},$$ $$S_{\widetilde W}=\int d^2x\,d\theta^+d\bar\theta^-\, \widetilde W(Y)+\text{c.c.}$$

The two holomorphic functions constrain different protected sectors. Mirror symmetry exchanges them rather than identifying them inside one description; a unified superspace treatment appears in Hori et al. 2003, ch. 11.

Vector multiplets and gauge-covariant chirality

For an Abelian vector superfield $V=V^\dagger$, the gauge transformation is

$$V\longmapsto V+\frac{i}{2}(\bar\Lambda-\Lambda), \qquad \bar D_\pm\Lambda=0.$$

The gauge-invariant field strength

$$\Sigma=\bar D_+D_-V$$

is twisted chiral: $\bar D_+\Sigma=D_-\Sigma=0$. Its lowest scalar $\sigma$, gauge field $A_\mu$, gauginos, and real auxiliary $D$ form the $(2,2)$ vector multiplet. The complexified FI parameter therefore appears in a twisted superpotential linear in $\Sigma$.

For non-Abelian $G$, $e^{2V}$ transforms by chiral group-valued factors and $\Sigma$ is adjoint-valued. Global data still matter: $SU(N)$, $PSU(N)$, and $U(N)$ have the same Lie algebra pieces but different allowed bundles, fluxes, theta angles, and line operators.

A controlled $(0,2)$ sector

The $(0,2)$ algebra retains $Q_+,\bar Q_+$ with $\{Q_+,\bar Q_+\}=2P_+$. In $(0,2)$ superspace, a chiral multiplet satisfies $\bar D_+\Phi=0$. A Fermi multiplet $\Gamma$ satisfies the deformed constraint

$$\bar D_+\Gamma=\sqrt2\,E(\Phi),$$

and can participate in a superpotential

$$S_J=-\frac1{\sqrt2}\int d^2x\,d\theta^+\, \Gamma_aJ^a(\Phi)+\text{c.c.}$$

Supersymmetry requires

$$\sum_a E_a(\Phi)J^a(\Phi)=0.$$

This follows by applying $\bar D_+$ to the integrand: the variation is proportional to $\sum_aE_aJ^a$. A $(2,2)$ chiral multiplet decomposes into one $(0,2)$ chiral and one Fermi multiplet with correlated $E$ and $J$ data. General $(0,2)$ theories relax those correlations and can have gauge, flavor, and gravitational anomalies; anomaly cancellation is not optional input.

Consistency checks

Before using a two-dimensional multiplet construction, check:

• signature and adjoints: Lorentzian Hermitian conjugation is what makes the positivity argument work;
• chirality convention: state whether $+$ labels spin or propagation direction and define $P_\pm$;
• reality: a complex Euclidean path integral treats barred fields as independent until a contour is chosen;
• R symmetry: distinguish classical assignments from the quantum non-anomalous subgroup;
• central sectors: a vanishing local central term does not rule out kink or winding charges;
• global gauge group: the Lie algebra alone does not determine fluxes and theta periodicity;
• reduction: identify which scalar came from a higher-dimensional gauge component rather than assuming every 2d multiplet has a four-dimensional ancestor.

Common pitfalls

Confusing chiral with right-moving. A chiral superfield is annihilated by both barred superspace derivatives. It contains fields of both Lorentz chiralities and is not a purely right-moving multiplet.

Treating $U(1)_A$ as automatically exact. In a charged theory its current can have a gauge anomaly proportional to the sum of matter charges. A B-twist or axial grading must therefore pass an anomaly check.

Ignoring the global group. Locally identical vector multiplet Lagrangians can define different theories when their flux lattices differ. This affects theta angles, orbifolds, and mirror maps.

Exercises

1. Verify directly from the displayed definitions that $\{D_+,\bar D_+\}=2i\partial_+$ and $D_+^2=0$.

Solution

Expand the two compositions on a test superfield. The two Grassmann derivatives cancel, as do the terms quadratic in $\bar\theta^+$; the cross terms add to $2i\partial_+$. Nilpotence follows because $(\partial_{\theta^+})^2=(\bar\theta^+)^2=0$ and their two cross terms have opposite signs.

2. Show that $\Sigma=\bar D_+D_-V$ is twisted chiral.

Solution

$\bar D_+\Sigma=\bar D_+^2D_-V=0$. Also $D_-\Sigma=-\bar D_+D_-^2V=0$, using $\{D_-,\bar D_+\}=0$.

3. Under the charge table above, verify that $\bar Q_++Q_-$ is neutral after twisting spin by $J_V/2$, while $\bar Q_++\bar Q_-$ is neutral after twisting by $J_A/2$.

Solution

The new spin is $s'=s+q_R/2$. For $\bar Q_+$, $(s,q_V,q_A)=(+\tfrac12,-1,-1)$, so either twist makes it scalar. For $Q_-$, $(s,q_V)=(-\tfrac12,+1)$, so the vector twist makes it scalar. For $\bar Q_-$, $(s,q_A)=(-\tfrac12,+1)$, so the axial twist makes it scalar.

References

• Gates, S. J., Hull, C. M., and Roček, M. “Twisted Multiplets and New Supersymmetric Nonlinear Sigma Models.” Nuclear Physics B 248 (1984): 157–186. doi:10.1016/0550-3213(84)90592-3.
• Hori, K., Katz, S., Klemm, A., Pandharipande, R., Thomas, R., Vafa, C., Vakil, R., and Zaslow, E. Mirror Symmetry. Clay Mathematics Monographs 1. Providence, RI: American Mathematical Society and Clay Mathematics Institute, 2003, chs. 11–12. Clay Mathematics Institute book page.