Rigid Supersymmetry from Nondynamical Supergravity Backgrounds

Rigid supersymmetry on a curved manifold is obtained by coupling the field theory to an off-shell supergravity multiplet, freezing all bosonic source fields, setting the gravitino to zero, and requiring its supersymmetry variation to vanish. Gravity is not quantized in this construction: the metric and auxiliary fields are nondynamical sources. The resulting generalized Killing-spinor equation, together with the matter transformations, defines the curved-space algebra.

Required background. Supercurrent multiplets and rigid-background compatibility determines which off-shell supergravity formulation the theory can couple to.

Helpful background. Spin structures and Dirac operators supplies the spin bundles in which the supersymmetry parameters live.

Freezing an off-shell supergravity multiplet

Let the supergravity fields be

$$\bigl(g_{\mu\nu},\psi_\mu;\,B_1,B_2,\ldots\bigr),$$

where $B_i$ are auxiliary or background gauge fields. The rigid limit consists of

$$\psi_\mu=0, \qquad g_{\mu\nu},B_i\ \text{fixed}, \qquad \delta_\varepsilon\psi_\mu(g,B_i)=0.$$

The last equation is differential in the spinor parameter $\varepsilon$ and algebraic in some auxiliary fields. Once it has a global solution, the supergravity transformation laws restricted to the frozen background give transformations of the matter multiplets. Off-shell closure matters: localization deforms the action far from the matter equations of motion, so an algebra that closes only on shell does not by itself control the functional integral.

This construction was formulated systematically by Festuccia and Seiberg 2011, §§2–3. It has three logically separate outputs:

1. the differential equation for the supersymmetry parameter;
2. the background-dependent matter transformations and curvature couplings;
3. the even symmetry generated by the square of a chosen odd transformation.

Solving only the first is not enough.

A four-dimensional new-minimal example

Consider a four-dimensional $\mathcal N=1$ theory with an R multiplet. On an oriented Riemannian four-manifold, new-minimal supergravity supplies an R connection $A_\mu^{(R)}$ and a conserved auxiliary vector $V_\mu$, with $\nabla_\mu V^\mu=0$. In the conventions of Dumitrescu, Festuccia, and Seiberg, the two chiral parameters satisfy

$$\begin{aligned} (\nabla_\mu-iA_\mu^{(R)})\zeta &=-iV_\mu\zeta-iV^\nu\sigma_{\mu\nu}\zeta,\\ (\nabla_\mu+iA_\mu^{(R)})\widetilde\zeta &=+iV_\mu\widetilde\zeta+iV^\nu\widetilde\sigma_{\mu\nu}\widetilde\zeta. \end{aligned}$$

Here $\zeta$ has R charge $+1$ and $\widetilde\zeta$ has R charge $-1$. This displayed equation fixes the convention: changing the sign of the R covariant derivative or defining $\sigma_{\mu\nu}$ with the opposite orientation changes corresponding signs everywhere else. The geometric consequences and the converse construction of the background fields are derived in Dumitrescu, Festuccia, and Seiberg 2012, §§1–3.

Euclidean reality requires special care. The spinors $\zeta$ and $\widetilde\zeta$ are independent complex sections; they are not related by Lorentzian Majorana conjugation. The background fields $A^{(R)}$ and $V$ may likewise be complex. A later choice of reflection structure or field-space integration cycle may impose a compatible real slice, but that choice is additional data. Festuccia and Seiberg give explicit examples where a supersymmetric Euclidean action exists without reflection positivity Festuccia and Seiberg 2011, §4.

Reading the rigid algebra

Suppose both $\zeta$ and $\widetilde\zeta$ are present and define the bilinear

$$K^\mu=\zeta\sigma^\mu\widetilde\zeta.$$

The Killing-spinor equations imply that $K$ is a complex Killing vector in this example. Acting on a field $\Phi$ of R charge $r$ and gauge representation $R_\Phi$, the mixed anticommutator has the structure

$$\{\delta_\zeta,\delta_{\widetilde\zeta}\}\Phi =2i\Bigl(\mathcal L_K\Phi -irK^\mu A_\mu^{(R)}\Phi +\delta_{\rm gauge}(\Lambda_K)\Phi +\cdots\Bigr).$$

The omitted terms are other background flavor transformations or local Lorentz rotations appropriate to the field. It is often convenient to combine them into a covariant Lie derivative $\mathcal L'_K$. The same-chirality anticommutators vanish in this new-minimal algebra. Exact factors of $i$ depend on whether gauge generators are Hermitian or anti-Hermitian; the invariant content is the full bosonic transformation generated by $Q^2$.

Before using this algebra, verify it on every multiplet, including auxiliary fields. For a localizing functional $V_{\rm loc}$ one needs

$$Q^2V_{\rm loc}=0,$$

which means invariance under the entire even transformation, not just under the spacetime isometry. The same requirement applies to insertions, gauge conditions, the field-space contour, and boundary conditions.

Integrability is necessary, not sufficient

Commuting two covariant derivatives in the generalized Killing-spinor equation relates the Riemann tensor and background fluxes. These integrability conditions efficiently rule out backgrounds, but they do not prove that a global spinor exists. One must still check transition functions, flux quantization, zeros of the spinor, and behavior at defects or boundaries. Those global questions are developed on the next page.

There is also no universal Killing-spinor equation independent of the supercurrent multiplet. Old-minimal supergravity uses different scalar and vector auxiliaries; extended supersymmetry brings additional R-symmetry bundles; conformal supergravity admits conformal Killing spinors. Translating between results therefore requires matching:

• signature and spinor conjugation;
• R-charge and gauge-covariant-derivative signs;
• normalization of auxiliary fields;
• orientation and gamma-matrix conventions;
• whether closure is off shell or uses equations of motion.

Boundaries and nondynamical gravity

On a manifold with boundary, a bulk solution of $\delta\psi_\mu=0$ is only the start. The vector generated by $Q^2$ must preserve the boundary, the variational principle must select $Q$-invariant boundary conditions, and normal supercurrent flux must either vanish or be cancelled by boundary degrees of freedom. Boundary counterterms can be forced by supersymmetry or anomaly inflow.

Finally, the frozen-source construction does not integrate over $g_{\mu\nu}$ or the auxiliaries and does not impose the supergravity equations of motion. A background may therefore be perfectly valid for rigid supersymmetry while failing to solve any dynamical gravity theory. Conversely, a supersymmetric supergravity solution may use a formulation to which the chosen QFT cannot couple.

Exercises

1. Flat-space limit. Set $A_\mu^{(R)}=V_\mu=0$ on flat $\mathbb R^4$. Solve the displayed new-minimal equations and identify the resulting bilinear.

Solution

Both equations reduce to $\partial_\mu\zeta=\partial_\mu\widetilde\zeta=0$, so the spinors are constant. Their bilinear $K^\mu=\zeta\sigma^\mu\widetilde\zeta$ is a constant translation vector, and the mixed anticommutator closes on translation plus any field-dependent gauge transformation in the gauge multiplet.

2. A forbidden deformation. Suppose $Q^2=\mathcal L_K+\delta_R(\alpha)$ and $V_{\rm loc}$ is invariant under $K$ but has nonzero R charge. Can $tQV_{\rm loc}$ be added while preserving $Q$?

Solution

No. Its $Q$ variation is $tQ^2V_{\rm loc}=t\delta_R(\alpha)V_{\rm loc}\ne0$. One must choose an R-neutral functional, or modify it so that it is invariant under the full even symmetry.

References

• Dumitrescu, Thomas T., Guido Festuccia, and Nathan Seiberg. “Exploring Curved Superspace.” Journal of High Energy Physics 2012, no. 8 (2012): 141. doi:10.1007/JHEP08(2012)141. 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.

Next step

Turn a local solution into a genuine symmetry by checking its global spin-R bundle and patching data.