Prepotentials for linearized supergravity

TL;DR

This paper reformulates linearized supergravity in any dimension using a first order formalism with prepotentials, enabling a geometric and conformally invariant approach to graviton and gravitino dynamics, including supersymmetry and dimensional reduction.

Contribution

It introduces a novel prepotential formalism for linearized supergravity in arbitrary dimensions, incorporating conformal invariance and duality, extending previous work to include gravitino fields and supersymmetry.

Findings

Prepotentials exhibit conformal invariance and simplify equations of motion.

Equations of motion are equivalent to twisted self-duality conditions.

Dimensional reduction reproduces known supergravity theories.

Abstract

Linearized supergravity in arbitrary dimension is reformulated into a first order formalism which treats the graviton and its dual on the same footing at the level of the action. This generalizes previous work by other authors in two directions: 1) we work in arbitrary space-time dimension, and 2) the gravitino field and supersymmetry are also considered. This requires the construction of conformally invariant curvatures (the Cotton fields) for a family of mixed symmetry tensors and tensor-spinors, whose properties we prove (invariance; completeness; conformal Poincar\'e lemma). We use these geometric tools to solve the Hamiltonian constraints appearing in the first order formalism of the graviton and gravitino: the constraints are solved through the introduction of prepotentials enjoying (linearized) conformal invariance. These new variables (two tensor fields for the graviton, one tensor-spinor for the gravitino) are injected into the action and equations of motion, which take a geometrically simple form in terms of the Cotton tensor(-spinors) of the prepotentials. In particular, the equations of motion of the graviton are equivalent to twisted self-duality conditions. We express the supersymmetric transformations of the graviton and gravitino into each other in terms of the prepotentials. We also reproduce the dimensional reduction of supergravity within the prepotential formalism. Finally, our formulas in dimension five are recovered from the dimensional reduction of the already known prepotential formulation of the six-dimensional $\mathcal{N}=(4,0)$ maximally supersymmetric theory.