Abstract
Starting with the light-cone Hamiltonian for gravity, we perform a field redefinition that reveals a hidden symmetry in four dimensions, namely the Ehlers $SL(2,R)$ symmetry. The field redefinition, which is non-local in space but local in time, acts as a canonical transformation in the Hamiltonian formulation keeping the Poisson bracket relations unaltered. We discuss the electro-magnetic duality symmetry of gravity in the light-cone formalism, which forms the $SO(2)$ subgroup of the Ehlers symmetry. The helicity states in the original Hamiltonian are not in a representation of the enhanced symmetry group. In order to make the symmetry manifest, we make a change of variables in the path integral from the helicity states to new fields that transform linearly under the $SO(2)$ duality symmetry.
Highlights
Theories of gravity and supergravity have been known to exhibit rich hidden symmetries upon dimensional reduction from their higher-dimensional parent theories
In [12], the Ehlers symmetry in four dimensions was obtained as a remnant of this E8 symmetry in N 1⁄4 8 supergravity after supersymmetric truncation to pure gravity
The obvious step would be to repeat the analysis presented here for the N 1⁄4 8 theory to find a suitable map from the helicity states to the appropriate field variables, on which the E8ð8Þ symmetry can be made manifest in four dimensions
Summary
Theories of gravity and supergravity have been known to exhibit rich hidden symmetries upon dimensional reduction from their higher-dimensional parent theories. Some of the celebrated examples include the Ehlers symmetry and the infinite-dimensional Geroch group in Einstein’s gravity [1,2] and the exceptional symmetries in maximal supergravity theories [3,4,5]. These hidden symmetries are believed to appear when the large spacetime symmetry group in the parent theory splits into a smaller one in lower dimensions, along with some internal symmetries that can further be enhanced using electromagnetic duality. We conclude with a few remarks about the relevance of our results to the more sophisticated frameworks, such as the prepotential formalism, exceptional field theories, etc., which explore these symmetry structures at greater lengths
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