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Abstract

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
fields.