Weyl and transverse diffeomorphism invariant spin-2 models in $$D=2+1$$ D = 2 + 1

Abstract

There are two covariant descriptions of massless spin-2 particles in D = 3 + 1 via a symmetric rank-2 tensor: the linearized Einstein-Hilbert (LEH) theory and the Weyl plus transverse diffeomorphism (WTDIFF) invariant model. From the LEH theory one can obtain the linearized new massive gravity (NMG) in D = 2 + 1 via Kaluza-Klein dimensional reduction followed by a dual master action. Here we show that a similar route takes us from the WTDIFF model to a linearized scalar-tensor NMG which belongs to a larger class of consistent spin-0 modifications of NMG. We also show that a traceless master action applied to a parity singlet furnishes two new spin-2 self-dual models. Moreover, we examine the singular replacement h μν → h μν − η μν h/D and prove that it leads to consistent massive spin-2 models in D = 2 + 1. They include linearized versions of unimodular topologically massive gravity (TMG) and unimodular NMG. Although the free part of those unimodular theories are Weyl invariant, we do not expect any improvement in the renormalizability. Both the linearized K-term (in NMG) and the linearized gravitational Chern-Simons term (in TMG) are invariant under longitudinal reparametrizations δh μν = ∂ μ ∂ ν ζ , which is not a symmetry of the WTDIFF Einstein-Hilbert term. Therefore, we still have one degree of freedom whose propagator behaves like 1/ p 2 for large momentum.

On higher spin analogues of linearized Topologically Massive Gravity and linearized "New Massive Gravity"

“…For instance, in the s = 2 case there are two choices for the D-tensor, one corresponds to the linearized Einstein-Hilbert theory and the other one to the WTDiff model or linearized unimodular gravity, both Lagrangians have no propagating modes in D = 2 + 1. The respective self-dual models L SD 2s−1 are the linearized topologically massive gravity [1] and linearized unimodular topologically massive gravity [23]. One can also combine L (s) 2s and L (s) 2s−2 and build up doublet models L D 2s containing both helicities +s and −s.…”

“…One can also combine L (s) 2s and L (s) 2s−2 and build up doublet models L D 2s containing both helicities +s and −s. They represent higher spin analogues of the linearized NMG [17] and of the linearized unimodular NMG [23]. In the next section, as a preparation for the section 4 where possible choices for D µ 1 µ 2 •••µs will be discussed, we give closed formulae for the Cotton tensor and its symmetrized curl on the flat space.…”

“…We can only have an empty spectrum if we enlarge the TDiff symmetry either to unconstrained linearized diffeomorphisms (Diff) by fixing 8 (a, b) = (1, 1) or to WTDiff (Weyl plus TDiff) by choosing (a, b) = (2/3, 5/9). The second case has been investigated in [23] and corresponds to the linearized version of unimodular gravity, its higher spin analogue will be investigated elsewhere [33] from the point of view of gauge invariants. Here we only work with the linearized Einstein-Hilbert theory L (1,1).…”

“…Consequently the third order self-dual model defined in (11) might contain a scalar ghost besides the helicity 2|m|/m particle. In order to avoid such extra modes we have learned from previous works [1,28,12,22,23] that L (s) 2s−2 must have an empty spectrum. In the spin-2 case this leads to only two possibilities.…”

“…Namely, either TDiff is extended to Diff (Einstein-Hilbert) or to WTDiff (linearized unimodular gravity). The corresponding self-dual models become the TMG of [1] and the unimodular TMG of [23] respectively.…”