Any ghost-free spin-2 or spin-3 TDiff model is equivalent, via source and field redefinitions, to the Maxwell-like model, which matches the string tensionless doublet action.
On the relation between local and geometric Lagrangians for higher spins
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abstract
Equations of motion for free higher-spin gauge fields of any symmetry can be formulated in terms of linearised curvatures. On the other hand, gauge invariance alone does not fix the form of the corresponding actions which, in addition, either contain higher derivatives or involve inverse powers of the d'Alembertian operator, thus introducing possible subtleties in degrees of freedom count. We suggest a path to avoid ambiguities, starting from local, unconstrained Lagrangians previously proposed, and integrating out the auxiliary fields from the functional integral, thus generating a unique non-local theory expressed in terms of curvatures.
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Equivalence of spin-2 and spin-3 models invariant under transverse diffeomorphisms and the tensionless limit of string theory
Any ghost-free spin-2 or spin-3 TDiff model is equivalent, via source and field redefinitions, to the Maxwell-like model, which matches the string tensionless doublet action.