On massive gravitons in 2+1 dimensions

TL;DR

This paper analyzes 3D theories of massive gravitons by comparing two covariant, parity-preserving extensions of the Fierz-Pauli spin-2 field: New Massive Gravity (NMG) and 3D bigravity. It shows that both reproduce FP linear dynamics and can be formulated with an auxiliary field to demonstrate on-shell and off-shell equivalence at the quadratic level, with explicit actions $S_{NMG}[g]$ and $S^{(2)}_{BT}[h,f]$ that realize the FP propagating modes. In 3D, the Einstein–Hilbert term propagates no massless modes, enabling a unitary massive spin-2 sector in NMG, while unitarity constrains parameters (e.g. $\alpha(\alpha+1)>0$) in bigravity and relates to the sign of curvature terms. The paper also discusses cosmological extensions to $AdS_3$, where the dual 2D CFT central charge scales as $c \propto (\alpha+1)$, revealing bulk/boundary unitarity trade-offs and placing NMG and bigravity within the broader landscape of 3D massive gravity models, including ties to TMG and GMG.

Abstract

The Fierz-Pauli (FP) free field theory for massive spin 2 particles can be extended, in a spacetime of (1+2) dimensions (3D), to a generally covariant parity-preserving interacting field theory, in at least two ways. One is "new massive gravity" (NMG), with an action that involves curvature-squared terms. Another is 3D "bigravity", which involves non-linear couplings of the FP tensor field to 3D Einstein-Hilbert gravity. We review the proof of the linearized equivalence of both "massive 3D gravity" theories to FP theory, and we comment on their similarities and differences.