Gravitational electric–magnetic duality, gauge invariance and twisted self-duality

Abstract

The concept of electric–magnetic duality can be extended to linearized gravity. It has indeed been established that in four dimensions, the Pauli–Fierz action (quadratic part of the Einstein–Hilbert action) can be cast in a form that is manifestly invariant under duality rotations in the internal 2-plane of the spacetime curvature and its dual. In order to achieve this manifestly duality-invariant form, it is necessary to introduce two ‘prepotentials’, which form a duality multiplet. These prepotentials enjoy interesting gauge invariance symmetries, which are, for each, linearized diffeomorphisms and linearized Weyl rescalings. The purpose of this note is twofold: (i) to rewrite the manifestly-duality invariant action obtained in previous work in a way that makes its gauge invariances also manifest, (ii) to explicitly show that the equations of motion derived from that action can be interpreted as twisted self-duality conditions on the curvature tensors of the two metrics obtained from the two prepotentials.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Higher spin theories and holography’.