The known ghost-free multivielbein theory is the unique ghost-free theory with genuine multi-field interactions for more than two vielbeins.
A note on "symmetric" vielbeins in bimetric, massive, perturbative and non perturbative gravities
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abstract
We consider a manifold endowed with two different vielbeins $E^{A}{}_{\mu}$ and $L^{A}{}_{\mu}$ corresponding to two different metrics $g_{\mu\nu}$ and $f_{\mu\nu}$. Such a situation arises generically in bimetric or massive gravity (including the recently discussed version of de Rham, Gabadadze and Tolley), as well as in perturbative quantum gravity where one vielbein parametrizes the background space-time and the other the dynamical degrees of freedom. We determine the conditions under which the relation $g^{\mu\nu} E^{A}{}_{\mu} L^{B}{}_{\nu} = g^{\mu\nu} E^{B}{}_{\mu} L^{A}{}_{\nu}$ can be imposed (or the "Deser-van Nieuwenhuizen" gauge chosen). We clarify and correct various statements which have been made about this issue.
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Bimetric interactions are defined via a congruence matrix, with the square root shown as the unique power series solution and algebraic equivalence to the unconstrained vielbein formulation.
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On the Uniqueness of Ghost-Free Multi-Gravity -- II: Constraining antisymmetrised multi spin-2 interactions
The known ghost-free multivielbein theory is the unique ghost-free theory with genuine multi-field interactions for more than two vielbeins.
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Bimetric interactions based on metric congruences
Bimetric interactions are defined via a congruence matrix, with the square root shown as the unique power series solution and algebraic equivalence to the unconstrained vielbein formulation.