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Journal Article

A note on the double dual graviton


Amaury,  Leonard
Quantum Gravity & Unified Theories, AEI-Golm, MPI for Gravitational Physics, Max Planck Society;

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Henneaux, M., Lekeu, V., & Amaury, L. (2020). A note on the double dual graviton. Journal of Physics A, 53(1): 014002. doi:10.1088/1751-8121/ab56ed.

Cite as: https://hdl.handle.net/21.11116/0000-0004-DB40-D
The (free) graviton admits, in addition to the standard Pauli-Fierz
description by means of a rank-two symmetric tensor, a description in which one
dualizes the corresponding (2,2)-curvature tensor on one column to get a
(D-2,2)-tensor, where D is the spacetime dimension. This tensor derives from a
gauge field with mixed Yound symmetry (D-3,1) called the "dual graviton" field.
The dual graviton field is related non-locally to the Pauli-Fierz field (even
on-shell), in much the same way as a p-form potential and its dual (D-p-2)-form
potential are related in the theory of an abelian p-form. Since the Pauli-Fierz
field has a Young tableau with two columns (of one box each), one can
contemplate a double dual description in which one dualizes on both columns and
not just on one. The double dual curvature is now a (D-2,D-2)-tensor and
derives from a gauge field with (D-3, D-3) mixed Young symmetry, the "double
dual graviton" field. We show, however, that the double dual graviton field is
algebraically and locally related to the original Pauli-Fierz field and, so,
does not provide a truly new description of the graviton. From this point of
view, it plays a very different role from the dual graviton field obtained
through a single dualization. We also show that these equations can be obtained
from a variational principle in which the variables to be varied in the action
are (all) the components of the double-dual field as well as an auxiliary field
with (2,1) Young symmetry. By gauge fixing the shift symmetries of this action
principle, one recovers the Pauli-Fierz action. Our approach differs from the
interesting approach based on parent actions and covers only the free,
sourceless theory. Similar results are argued to hold for higher spin gauge