Duality in linearized gravity

TL;DR

This work demonstrates that $SO(2)$ duality rotations of the linearized gravitational curvature can be extended to a symmetry of the Pauli-Fierz action in four dimensions by introducing two local symmetric two-index superpotentials. The duality acts as an $SO(2)$ rotation on the pair of potentials $(P_{mn},\\Phi_{mn})$, yielding a Maxwell-like action with a conserved Noether charge that has a Chern-Simons-like form. The authors derive explicit transformation rules, prove invariance of the Hamiltonian and kinetic terms, and present a manifestly duality-invariant $SO(2)$-vector formulation. These results deepen the understanding of gravitational duality in the linear regime and hint at connections to hidden symmetries in dimensional reduction and potential extensions to non-linear gravity.

Abstract

We show that duality transformations of linearized gravity in four dimensions, i.e., rotations of the linearized Riemann tensor and its dual into each other, can be extended to the dynamical fields of the theory so as to be symmetries of the action and not just symmetries of the equations of motion. Our approach relies on the introduction of two "superpotentials", one for the spatial components of the spin-2 field and the other for their canonically conjugate momenta. These superpotentials are two-index, symmetric tensors. They can be taken to be the basic dynamical fields and appear locally in the action. They are simply rotated into each other under duality. In terms of the superpotentials, the canonical generator of duality rotations is found to have a Chern-Simons like structure, as in the Maxwell case.