Metric-Like Formalism for Matter Fields Coupled to 3D Higher Spin Gravity

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 first in the frame-like formalism. The field $C$ satisfies an eq of motion, $\partial_{\mu} \, C+A_{\mu} \, C-C \, \bar{A}_{\mu}=0$. With a suitable gauge fixing of a new local symmetry and diffeomorphism, only one component of $B_{\mu\nu}$, say $B_{\phi r}$, remains non-vanishing and satisfies an equation similar to that for $C$ with $A_{\mu}$ and $\bar{A}_{\mu}$ interchanged. The spin connection is eliminated by solving the eq of motion for the total action, and 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]. The 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 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 re-derived. Then the matter action is re-expressed in terms of $g_{\mu\nu}$, $\phi_{\mu\nu\rho}$ and the covariant derivatives for spin-3 geometry. Spin-3 gauge transformation is extended to the matter fields.