Supercurrent Multiplets and Rigid-Background Compatibility

A curved supersymmetric background can source only a current multiplet that exists globally in the quantum field theory. In four-dimensional $\mathcal N=1$ language, the universal $S$-multiplet can sometimes be improved to the Ferrara–Zumino or R multiplet; that improvement determines whether old-minimal, new-minimal, or a larger off-shell supergravity formulation can be used. The choice must be made before freezing any background fields.

Required background. Supercurrent multiplets and improvements supplies the superspace conservation equations. Current sources and generating functionals supplies the source–operator pairing and its contact terms.

Helpful background. What is an anomaly? distinguishes a genuine obstruction to gauging a background symmetry from a removable local contact term.

The source multiplet determines the background

Work in flat four-dimensional $\mathcal N=1$ superspace while identifying the sources. In one standard normalization the real superfield $\mathcal S_{\alpha\dot\alpha}$ obeys

$$\bar D^{\dot\alpha}\mathcal S_{\alpha\dot\alpha} =D_\alpha X+\chi_\alpha, \qquad \bar D_{\dot\alpha}X=0, \qquad \bar D_{\dot\alpha}\chi_\alpha=0,$$

together with $\bar D_{\dot\alpha}\bar\chi^{\dot\alpha}=D^\alpha\chi_\alpha$. Its components include $T_{\mu\nu}$, the supersymmetry current, a vector current, and currents carried by strings or domain walls. A linear source coupling has the schematic form

$$S_{\rm source}=\int d^4x\,d^4\theta\; H^{\alpha\dot\alpha}\mathcal S_{\alpha\dot\alpha} +\text{compensator couplings},$$

so the conservation equation is exactly the condition for invariance under the linearized supergravity gauge transformations. This is the supersymmetric version of $\int h_{\mu\nu}T^{\mu\nu}$: changing the current multiplet changes the compensators and auxiliary sources, not merely notation. The classification and its supergravity couplings are derived in Komargodski and Seiberg 2010, §§2–5.

For a real superfield $U$, an improvement can be written in this normalization as

$$\begin{aligned} \mathcal S_{\alpha\dot\alpha}&\longmapsto \mathcal S_{\alpha\dot\alpha}+[D_\alpha,\bar D_{\dot\alpha}]U,\\ X&\longmapsto X+\tfrac12\bar D^2U,\\ \chi_\alpha&\longmapsto \chi_\alpha+\tfrac32\bar D^2D_\alpha U. \end{aligned}$$

Numerical coefficients change if $X$ and $\chi_\alpha$ are rescaled, but the two invariant questions do not:

• Can one choose a globally defined, gauge-invariant $U$ for which $\chi_\alpha=0$? Then the Ferrara–Zumino (FZ) multiplet exists and couples to old-minimal supergravity.
• Can one choose such a $U$ for which $X=0$? Then the R multiplet exists, its bottom component is a conserved nonanomalous $U(1)_R$ current, and it couples to new-minimal supergravity.

When neither reduction is available, the $S$-multiplet can couple at the linearized level to a larger supergravity multiplet with additional compensating degrees of freedom. It is not consistent to select the auxiliary fields of old- or new-minimal supergravity while ignoring the missing improvement.

Global obstructions to an improvement

An improvement that works in one coordinate patch may fail as a source coupling. Two standard failures are physical:

A Fayet–Iliopoulos term. For a dynamical Abelian vector multiplet, the candidate FZ improvement is not gauge invariant. The $S$-multiplet remains well defined, but a gauge-invariant FZ multiplet need not exist.

A non-exact Kähler form. In a nonlinear sigma model the local Kähler potential changes by $K_i-K_j=f_{ij}+\bar f_{ij}$ between target-space patches. If the Kähler form represents a nontrivial cohomology class, an improvement built from $K$ does not patch to a global real superfield. Thus the local FZ expression is not a global operator. These obstructions, and the fact that the $S$-multiplet retains the associated brane currents, are established in Komargodski and Seiberg 2010, §§3–4 and generalized across dimensions in Dumitrescu and Seiberg 2011, §§2–4.

The R-multiplet has a different obstruction: the theory must possess an exact continuous $U(1)_R$ symmetry. A classical charge assignment is insufficient if the current has an ABJ anomaly, is explicitly broken by the superpotential, or exists only after mixing with a flavor current that is not globally gaugeable. Background ’t Hooft anomalies do not necessarily destroy the current, but they do control non-invariance of the generating functional and must be represented by inflow or contact terms.

A compatibility test

Given a proposed rigid background, proceed in this order:

1. Name the off-shell formulation. List its metric, R gauge field, tensor or vector sources, scalar auxiliaries, and compensator.
2. Match each source to an operator. Verify that the operators sit in one conserved supercurrent multiplet.
3. Construct the required improvement. Check it on overlaps in target space and spacetime, and under dynamical gauge transformations.
4. Check the quantum symmetry. Include ABJ anomalies and allowed local contact terms.
5. Only then freeze the sources. The gravitino equation of that same formulation defines the admissible rigid geometries.

For example, a theory with a nonanomalous $U(1)_R$ current but no global FZ multiplet may be placed on a new-minimal background; lack of an FZ multiplet is not an obstruction there. Conversely, a theory without a continuous R symmetry cannot simply borrow the new-minimal Killing-spinor equation. It may admit an old-minimal background if the FZ multiplet exists, but the allowed geometries and reflection properties differ. Festuccia and Seiberg’s frozen-supergravity construction makes this dependence explicit Festuccia and Seiberg 2011, §§2–3.

Quantum qualifications

The supercurrent equation is an operator statement. Renormalization can mix improvements with local curvature terms, and the generating functional may shift under a change of scheme by a local supersymmetric functional of background fields. Such a shift can change contact terms or a curved-space partition function without changing separated flat-space correlators. Therefore “supersymmetric background exists” and “its partition function is scheme independent” are separate conclusions. A systematic four-dimensional new-minimal counterterm classification is given by Assel, Cassani, and Martelli 2014, §§3–5.

Exercises

1. Why does a local Kähler potential not suffice? Let $K_i-K_j=f_{ij}+\bar f_{ij}$ on overlapping target-space charts. Explain why an improvement $U_i\propto K_i$ does not define a global real superfield when the Kähler class is nonzero.

Solution

On an overlap, $U_i-U_j\propto f_{ij}+\bar f_{ij}$. Although this is a Kähler transformation of the sigma-model action, it is a nonzero transition for the proposed operator $U$. The improved supercurrent therefore differs from patch to patch. A global improvement would make the Kähler form exact; a nonzero Kähler class forbids it.

2. Choose the formulation. A theory has a gauge-invariant FZ multiplet and an anomaly-free $U(1)_R$ current. Which minimal formulations are available, and is their rigid-background problem identical?

Solution

Both old-minimal and new-minimal couplings are available. They use different compensators and auxiliary fields, so their gravitino variations and allowed frozen backgrounds are not identical. Any comparison must translate the background sources and the improvement relating their current multiplets.

References

• Assel, Benjamin, Davide Cassani, and Dario Martelli. “Supersymmetric Counterterms from New Minimal Supergravity.” Journal of High Energy Physics 2014, no. 11 (2014): 135. doi:10.1007/JHEP11(2014)135. Open preprint.
• Dumitrescu, Thomas T., and Nathan Seiberg. “Supercurrents and Brane Currents in Diverse Dimensions.” Journal of High Energy Physics 2011, no. 7 (2011): 095. doi:10.1007/JHEP07(2011)095. Open preprint.
• Festuccia, Guido, and Nathan Seiberg. “Rigid Supersymmetric Theories in Curved Superspace.” Journal of High Energy Physics 2011, no. 6 (2011): 114. doi:10.1007/JHEP06(2011)114. Open preprint.
• Komargodski, Zohar, and Nathan Seiberg. “Comments on Supercurrent Multiplets, Supersymmetric Field Theories and Supergravity.” Journal of High Energy Physics 2010, no. 7 (2010): 017. doi:10.1007/JHEP07(2010)017. Open preprint.

Next step

With the admissible source multiplet fixed, derive rigid supersymmetry from its nondynamical supergravity background.