In $D=2+1$ dimensions there are two dual descriptions of parity singlets of helicity $\pm 1$, namely, the self-dual model of first order (in derivatives) and the Maxwell-Chern-Simons theory of second order. Correspondingly, for helicity $\pm 2$ there are four models $S_{\mathrm{SD}\pm }^{(r)}$ describing parity singlets of helicities $\pm 2$. 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 $S_{\mathrm{SD}+}^{(4)}$ and $S_{\mathrm{SD}-}^{(4)}$ leads to the linearized form of the new massive gravity suggested by Bergshoeff, Hohm, and Townsend (BHT) similarly to the soldering of $S_{\mathrm{SD}+}^{(3)}$ and $S_{\mathrm{SD}-}^{(3)}$. 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, $S_{\mathrm{SD}+}^{(3)}+S_{\mathrm{SD}-}^{(3)}$, and $S_{\mathrm{SD}+}^{(4)}+S_{\mathrm{SD}-}^{(4)}$. 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.

• Received 14 August 2010

DOI:https://doi.org/10.1103/PhysRevD.82.105009