There are three self-dual models of massive particles of helicity +2 (or −2) in D = 2+ 1. Each model is of first, second, and third-order in derivatives. Here we derive a new self-dual model of fourth-order, L (4) SD are encoded in a ranktwo tensor which is symmetric, traceless and transverse due to trivial (non-dynamic) identities, contrary to other spin-2 self-dual models. We also show that the Noether embedment of the Fierz-Pauli theory leads to the new massive gravity of Bergshoeff, Hohm and Townsend.

The direct sum of a couple of Maxwell-Chern-Simons (MCS) gauge theories of opposite helicities ±1 does not lead to a Proca theory in D = 2 + 1, although both theories share the same spectrum. However, it is known that by adding an interference term between both helicities we can join the complementary pieces together and obtain the physically expected result. A generalized soldering procedure can be defined to generate the missing interference term. Here we show that the same procedure can be applied to join together ±2 helicity states in a full off-shell manner. In particular, by using second-order (in derivatives) self-dual models of helicities ±2 (spin two analogues of MCS models) the Fierz-Pauli theory is obtained after soldering. Remarkably, if we replace the second-order models by third-order self-dual models (linearized topologically massive gravity) of opposite helicities we end up after soldering exactly with the new massive gravity theory of Bergshoeff, Hohm and Townsend in its linearized approximation.

We present here a relationship among massive self-dual models for spin-3 particles in D = 2 + 1 via the Noether Gauge Embedment (N GE) procedure. Starting with a first order model (in derivatives) S SD(1) we have obtained a sequence of four self-dual models S SD(i) where i = 1, 2, 3, 4. We demonstrate that the N GE procedure generate the correct action for the auxiliary fields automatically. We obtain the whole action for the 4th order self-dual model including all the needed auxiliary fields to get rid of the ghosts of the theory.

In D = 2 + 1 dimensions there are two dual descriptions of parity singlets of helicity ±1, namely the self-dual model of first-order (in derivatives) and the Maxwell-Chern-Simons theory of second-order. Correspondingly, for helicity ±2 there are four models S (r) SD± describing parity singlets of helicities ±2. They are of first-, second-,third-and fourth-order (r = 1, 2, 3, 4) respectively. Here we show that the generalized soldering of the opposite helicity models S (4) SD+ and S (4) SD− leads to the linearized form of the new massive gravity suggested by Bergshoeff, Hohm and Townsend (BHT) similarly to the soldering of S (3) SD+ and S (3)SD− . We argue why in both cases we have the same result. We also find out a triple master action which interpolates between the three dual models: linearized BHT theory, S (3) SD+ + S (3) SD− and S (4) SD+ + S (4)SD− . By comparing gauge invariant correlation functions we deduce dual maps between those models. In particular, we learn how to decompose the field of the linearized BHT theory in helicity eigenstates of the dual models up to gauge transformations.

We find new higher derivative models describing a parity doublet of massive spin-3 modes in D ¼ 2 þ 1 dimensions. One of them is of fourth order in derivatives while the other one is of sixth order. They are complete, in the sense that they contain the auxiliary scalar field required to remove spurious degrees of freedom. Both of them are obtained through the master action technique starting with the usual (second-order) spin-3 Singh-Hagen model, which guarantees that they are ghost free. The fourth-and sixthorder terms are both invariant under (transverse) Weyl transformations, quite similarly to the fourth-order K-term of the "new massive gravity." The sixth-order term slightly differs from the product of the Schouten by the Einstein tensor, both of third order in derivatives. It is also possible to write down the fourth-order term as a product of a Schouten-like by an Einstein-like tensor (both of second order in derivatives) in close analogy with the K-term.