Rigid Supersymmetry on Curved Spaces

A supersymmetric theory in flat space does not automatically remain supersymmetric when its derivatives are covariantized. The metric changes the spinor derivative, and the supersymmetry parameter need no longer be parallel. This volume explains how extra fixed sources and their transformation laws organize a consistent curved-space theory.

The first goal is concrete: specify a background and test whether it preserves a nonzero supersymmetry parameter. The gravitational fields serve as sources for the matter theory. We do not integrate over metrics or impose gravitational equations of motion.

Start with a variation that fails

Why minimal coupling is not enough varies the free chiral action with a precisely stated naive covariantization. It exposes derivatives of the spinor label in the bulk variation and separately checks the curvature obstruction to a parallel spinor. An explicit curved metric and Gaussian matter profile give a nonzero integrated variation.

Prepare with R-symmetry and flavor symmetry and Covariant derivatives of spinors. The flat chiral calculation supplies the field transformations; the geometric lesson explains why constant frame components and a parallel spinor are different claims.

The useful question at the end is not merely whether the metric is curved. It is which terms in the action, transformation rules, and background equation must change together.

Freeze a set of sources

The R-multiplet through its physical currents explains why these probes belong together. Begin with a current/source dictionary, multiply the source and current superfields, and derive the R-current’s precise source. A free chiral example tracks how a flavor improvement changes the stress tensor and central-charge partners along with the R-current.

Supergravity as a system of background sources starts from a source-dependent action. A source for the supersymmetry current is fermionic, and its variation matters even when its value is zero. Freezing the sources retains precisely the transformations that preserve their chosen values.

For this step, Sources, currents, and generating functionals supplies the necessary functional-variation language. A finite Gaussian then makes the distinction between a fixed auxiliary source and an auxiliary variable integrated out of the theory explicit: they give different means and fluctuations.

Next, What each background field does identifies the metric, R-connection, dual vector, and scalar. Work through the local potential example and distinguish the connection in the matter derivative from the source coupled to the R-current. Their different coefficients are essential when translating between a transformation law and a source response.

Test a supersymmetry parameter

Derive and interpret the Killing spinor equations begins with the supplied gravitino variation and sets it to zero. It keeps the two opposite R-charge equations distinct, checks local gauge covariance, and solves a flat background whose spinor components vary with position.

You can follow this calculation without first reconstructing the full R-multiplet. The current-multiplet lesson prepares a later return to the origin of the supplied variation. The first route instead concentrates on using the equation correctly.

A successful local solution is only part of the answer. The circle exercise checks whether it satisfies the global spinor boundary condition, and shows how an R-connection can compensate an antiperiodic spin structure. The global gauge period is stated explicitly rather than inferred from a local differential equation.

Solve the equation on the round three-sphere uses the global sphere frame to check every spinor component. It also supplies regular local potentials for the background three-form and checks the global solutions. Continue to the curved-action volume for the matter transformations and complete action variation.

Check integrability and verify a proposed background derives the curvature of the complete spinor connection. A worked source repair shows why a common curvature kernel, a preserved line and a global solution are separate requirements. A flat quotient then exposes an obstruction that local curvature cannot detect.

The curved supersymmetry algebra then composes two transformations on an unconstrained charged scalar. It separates geometric transport, spin rotation and the internal source transformations, and tests their cancellation in a varying background.

Recover the linearized flat-space description checks the first variation against flat-space currents, derives one Q-exact deformation in components, and explains the extra information needed for a claim about a finite family.

Choose the depth needed for your calculation

The twenty-lesson round-sphere route selects the preparation for a first complete curved chiral theory. It does not require every branch of this volume. The extended route through the original paper also develops the R-multiplet, the curved algebra, the linearized consistency check, and more general integrability conditions.

As you work, keep three questions separate: which sources define the background, which global spinors solve its preservation equations, and which completed matter action realizes those symmetries. An answer to one gives input to the others rather than replacing their checks.