Abstract
Using the frame formulation of multi-gravity in three dimensions, we show that demanding the presence of secondary constraints which remove the Boulware-Deser ghosts restricts the possible interaction terms of the theory and identifies invertible frame field combinations whose effective metric may consistently couple to matter. The resulting ghost-free theories can be represented by theory graphs which are trees. In the case of three frame fields, we explicitly show that the requirement of positive masses and energies for the bulk spin-2 modes in AdS$_3$ is consistent with a positive central charge for the putative dual CFT$_2$.
Highlights
Theories considered in [10] possesses the necessary constraints needed to remove the additional scalar modes
Using the frame formulation of multi-gravity in three dimensions, we show that demanding the presence of secondary constraints which remove the Boulware-Deser ghosts restricts the possible interaction terms of the theory and identifies invertible frame field combinations whose effective metric may consistently couple to matter
In the case of three frame fields, we explicitly show that the requirement of positive masses and energies for the bulk spin-2 modes in AdS3 is consistent with a positive central charge for the putative dual CFT2
Summary
The frame formulation of multi-metric gravity was introduced in [10]. In this paper we restrict our attention to the three dimensional case. According to Dirac’s procedure for constraint Hamiltonian systems, additional constraints can follow from demanding that the primary constraints are conserved under time evolution This leads to a set of consistency conditions which in the case of Chern-Simons-like theories can equivalently be derived on-shell by using the Bianchi and Cartan identities satisfied by the curvature and torsion two-forms (see [21] for more details), DI RI = 0 , DI TI = RI × eI. If one restricts the coupling constants of the theory as β1β2 = 0, but one of them non-zero, the invertibility of a single dreibein and not of any linear combination is sufficient to define a ghost-free theory We can depict these interaction terms as arrows between nodes, e1. The three curvature two-forms satisfy three Bianchi identities (2.5), and the three torsions satisfy three Cartan identities (2.6)
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