Generalized soldering of $\pm 2$ helicity states in $D=2+1$

TL;DR

This paper extends the soldering technique to combine opposite helicity spin-2 states in 2+1 dimensions, deriving known massive gravity theories from self-dual models through an off-shell procedure.

Contribution

It introduces a generalized soldering method for spin-2 helicity states, connecting second- and third-order self-dual models to established massive gravity theories.

Findings

Soldering of second-order models yields Fierz-Pauli theory.

Soldering of third-order models reproduces linearized new massive gravity.

The method provides a unified off-shell framework for spin-2 helicity states.

Abstract

The direct sum of a couple of Maxwell-Chern-Simons (MCS) gauge theories of opposite helicities $\pm 1$ 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 $\pm 2$ helicity states in a full off-shell manner. In particular, by using second-order (in derivatives) self-dual models of helicities $\pm 2$ (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.