A note on the double dual graviton

Abstract

The (free) graviton admits, in addition to the standard Pauli-Fierz description by means of a ranktwo 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.

Duality and Higher Buscher Rules in p‐Form Gauge Theory and Linearized Gravity

“…Note also that in the absence of θ, the duality relation () becomes rather trivial, since the two would‐be dual gravitons are related algebraically rather than via a non‐local relation. This was observed and studied in [18]. However, a side result of our analysis is that in the presence of such a theta term, double duality does not lead to an algebraic relation between the two dual fields.…”

“…We demonstrate that this can also be implemented off‐shell and discuss what is the meaning of the θ‐term in that case. Second, we discuss the “double” duality of gravitons in the presence of θ‐term and comment on how the results of [18] can be modified in that case. In the spirit of completeness, we start with a short review of graded geometry and how Fierz‐Pauli action looks like in this formulation.…”

Graded Geometry and Tensor Gauge Theories

“…A first step towards this direction was made in [15], where all possible duals of differential forms were analysed on-shell and it was found that only a fraction of them are algebraically independent. This analysis is based on the preceding one for gravitons, which was performed in [16] and led to the result that the double dual of the graviton is not independent to the original field.…”

“…In several other terms the inverse boxes will cancel as desired, but there will still be terms where the inverse boxes do not cancel. These will be of the general form 16) namely ones with multiple cross-contracted derivatives. Note also that this issue cannot be ameliorated by partial integration.…”

Hypergravity in five dimensions

“…Analogously, the one-form e ab decomposes as e ab μ ∶fe ðμνλÞ ; e ν νμ ; e μν;λ g; ð31Þ so that its trace e ν νμ corresponds to a spin-1 field, while the fully symmetric part e ðμνλÞ describes the highest spin field (s ¼ 3). The remaining field e μν;λ is of mixed (2,1) Young symmetry, remarkably coinciding with the dual of the graviton in five dimensions [77] (see also [78]). It is worth highlighting that in our context, the field e μν;λ is strictly required by hypersymmetry, instead of duality at the linearized level.…”