Ippei Fujisawa, Ryuichi Nakayama
Action integral for a matter system composed of 0- and 2-forms, $C$ and $B_{\mu\nu}$, topologically coupled to 3D spin-3 gravity is considered in the frame-like formalism, and then the spin connection is eliminated by solving the eq of motion for the total action. It is shown that in the resulting metric-like formalism, $(BC)^2$ interaction terms are induced because of the torsion. The world-volume components of the matter field, $C^0$, $C^{\mu}$ and $C^{(\mu\nu)}$, are introduced by contracting the local-frame index of $C$ with those of the inverse vielbeins, $E_a^{\mu}$ and $E_a^{(\mu\nu)}$, which were defined by the present authors in ArXiv:1209.0894 [hep-th]. 3D higher spin gravity theory contains various metric-like fields. These metric-like fields, as well as the new connections and the generalized curvature tensors, introduced in the above mentioned paper, are explicitly expressed in terms of the metric $g_{\mu\nu}$ and the spin-3 field $\phi_{\mu\nu\lambda}$ by means of the $\phi$-expansion. The matter action is re-expressed in terms of $g_{\mu\nu}$, $\phi_{\mu\nu\rho}$ and the covariant derivatives for spin-3 geometry. It is found that the action in the metric-like formalism is invariant under diffeomorphisms, just because the action in the frame-like formalism is topological: the SL(3,R) $\times$ SL(3,R) gauge symmetry in the matter-coupled theory does not contain true diffeomorphisms. This is due to the extra terms in the transformation rules which depend on the matter fields. Spin-3 gauge transformation is extended to the matter fields. The action integral for the pure spin-3 gravity in the metric-like formalism up to ${\cal O}(\phi^2)$, obtained before in the literature, is rederived.
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http://arxiv.org/abs/1304.7941
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