Abstract
Starting from the dual Lagrangians recently obtained for (partially) massless spin-2 fields in the Stueckelberg formulation, we write the equations of motion for (partially) massless gravitons in (A)dS in the form of twisted-duality relations. In both cases, the latter admit a smooth flat limit. In the massless case, this limit reproduces the gravitational twisted-duality relations previously known for Minkowski spacetime. In the partially-massless case, our twisted-duality relations preserve the number of degrees of freedom in the flat limit, in the sense that they split into a decoupled pair of dualities for spin-1 and spin-2 fields. Our results apply to spacetimes of any dimension greater than three. In four dimensions, the twisted-duality relations for partially massless fields that appeared in the literature are recovered by gauging away the Stueckelberg field.
Highlights
AND CONVENTIONSElectric-magnetic duality, the symmetry of vacuum Maxwell equations under the exchange of electric and magnetic fields that interchanges dynamical equations with Bianchi identities, has counterparts in other physical systems, including supersymmetric field theories, linearised gravity and free higher-spin gauge theories
For linearised gravity on a flat background in n dimensions, for instance, it relates the Fierz-Pauli description in terms of the tensor hab with a description in terms of an irreducible mixed-symmetry tensor Ta1...an−3|b, completely antisymmetric in its first n − 3 indices. This link has been established only at the linearised level: non-linear Einstein gravity cannot be reproduced in the dual mixed-symmetry picture by means of local interactions [4, 5]
The problems encountered in the attempts to lift gravitational dualities from the linearised formulation in flat background to the interacting level suggest that the study of electric-magnetic duality in curved backgrounds may be promising
Summary
Electric-magnetic duality, the symmetry of vacuum Maxwell equations under the exchange of electric and magnetic fields that interchanges dynamical equations with Bianchi identities, has counterparts in other physical systems, including supersymmetric field theories, linearised gravity and free higher-spin gauge theories. We show that the field equations of two dual theories are formulated as a twisted-duality equation, we note that the latter is not obtained from a variational principle that is manifestly spacetime covariant. Forgoing the latter requirement, for linearised Einstein theory around flat spacetime Bunster et al [25] gave an action principle that yields the twisted self-duality conditions as equations of motion, keeping the graviton and its dual on equal footing. The symbols ǫa1···an and ǫa1···an denote the totally antisymmetric tensors obtained from t√he−cgo.rresponding densities upon multiplication and division by
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