Supersymmetric Actions on Curved Spaces

A background with a Killing spinor gives us a candidate supersymmetry parameter. To define a field theory, we must also specify how matter transforms and which action those transformations preserve. This volume connects those three ingredients through calculations in three-dimensional Euclidean field theory.

The first example is a gauge-neutral chiral multiplet on the round three-sphere. It is enough to follow the twenty-lesson route; the systematic construction of general interacting actions can be studied afterward.

Begin with the curved multiplet

The curved chiral multiplet starts with the familiar scalar, two-component fermion, and auxiliary field. Their R-charges differ, so each component needs its own charge in the background covariant derivative. The extra algebraic terms and the derivative of a complete spinor bilinear are essential to the curved transformation rules.

The preparation is R-symmetry and flavor symmetry together with the Killing spinor equations. The calculation introduces the identities needed to check closure on the sphere; it does not require a separate course in the general curved algebra.

Keep the distinction between a matter auxiliary field and an auxiliary background explicit. The former belongs to the off-shell matter multiplet. The latter is prescribed by the background construction. Eliminating one supplies no equation for the other.

Put the free theory on the sphere

For the geometry, Solve the equation on the round three-sphere specifies a global frame, the metric radius, all sources, and the preserved spinors. It also checks that their global behavior is consistent.

Then The free chiral theory on the round sphere supplies the complete action. Follow its variation term by term: the scalar kinetic term, fermion kinetic term, matter auxiliary term, scalar curvature-dependent coupling, and fermion algebraic coupling must work together.

The scalar R-charge remains explicit, so changing it tests the coefficients. The radius provides a second diagnostic: at fixed physical distances the additional terms vanish in the large-radius limit, recovering the flat action. Neither check replaces the full variation.

The sphere has no boundary. Once the local variation is a divergence of a globally defined current, its integral vanishes for smooth fields. This is the classical action check. A quantum functional integral also needs a contour, a regulator, and a treatment of any zero modes; those assumptions should be stated when that integral becomes the observable.

Continue to interactions and gauge fields

Build invariant F-terms and D-terms multiplies two curved chirals, checks the total charges needed for an invariant F-term, and derives the background completion of a neutral D-term. A concrete matter product recovers the sphere curvature coefficients.

The curved vector multiplet derives the scalar and gauge-potential algebra with nonconstant Killing labels. Calculate the shifted auxiliary field required by an invariant sphere source, and distinguish that source condition from a dynamical field equation. These ingredients prepare the gauge-matter action and the interpretation of real-mass deformations.

Gauge and matter actions from multiplets expands the charged-chiral product and derives its gauge, scalar, Yukawa and auxiliary couplings. Build the vector kinetic term from its field-strength multiplet, then check the flat and round-sphere limits with the same conventions.

Real masses and flavor backgrounds freezes the complete flavor multiplet. Derive the signed mass, the sphere auxiliary value and the insertion obtained by differentiating a supersymmetric source family. A compact-circle example separates the mass from a global flavor holonomy.

Chern–Simons and Fayet–Iliopoulos terms form another branch. The flat-space entrance derives their supersymmetric partners and a common topological mass; the curved completion uses the additional multiplet preparation. A choice of classical coupling and a statement about a regulated quantum phase answer different questions.

At the end of an action calculation, be able to identify every field and coefficient, reproduce the supersymmetry variation, and say which boundary conditions were used. The rigid-background overview is the place to return when the missing information concerns the sources or the preserved spinor rather than the matter action.