Abstract
We derive a manifestly duality-symmetric formulation of the action principle for conformal gravity linearized around Minkowski space-time. The analysis is performed in the Hamiltonian formulation, the fourth-order character of the equations of motion requiring the formal treatment of the three-dimensional metric perturbation and the extrinsic curvature as independent dynamical variables. The constraints are solved in terms of two symmetric potentials that are interpreted as a dual three-dimensional metric and a dual extrinsic curvature. The action principle can be written in terms of these four dynamical variables, duality acting as simultaneous rotations in the respective spaces spanned by the three-dimensional metrics and the extrinsic curvatures. A twisted self-duality formulation of the equations of motion is also provided.
Highlights
Understanding dualities is a major challenge in modern theoretical physics
A generalization of electric-magnetic duality in conformal gravity was studied in the early work [22], where the Euclidean action with a gauge-fixed metric was expressed in terms of quadratic forms involving the electric and magnetic components of the Weyl tensor, exhibiting a discrete duality symmetry upon the interchange of these components
We have shown that electric-magnetic duality is a hidden symmetry of linearized conformal gravity, both at the level of the equations of motion and the action principle
Summary
Understanding dualities is a major challenge in modern theoretical physics. Despite their widespread presence in field theory, (super)gravity and string theory, the current understanding of their origins and full implications is rather limited. The graviton and its dual are each described by a symmetric rank-two tensor field, and a duality symmetry relating them is expected to emerge, inherited from the underlying infinite-dimensional algebraic structure This has motivated the search of duality-symmetric action principles involving gravity. A generalization of electric-magnetic duality in conformal gravity was studied in the early work [22], where the Euclidean action with a gauge-fixed metric was expressed in terms of quadratic forms involving the electric and magnetic components of the Weyl tensor, exhibiting a discrete duality symmetry upon the interchange of these components. The structure of the duality-symmetric action principle is new, different from duality-invariant Maxwell theory and linearized gravity: duality acts rotating simultaneously the three-dimensional metrics ðhij; hijÞ and the extrinsic curvatures ðKij; KijÞ.
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