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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 $B_{μν}$, topologically coupled to 3D spin-3 gravity is considered first in the frame-like formalism. The field $C$ satisfies an eq of motion, $\partial_μ \, C+A_μ \, C-C \, \bar{A}_μ=0$. With a suitable gauge fixing of a new local symmetry and diffeomorphism, only one component of $B_{μν}$, say $B_{ϕr}$, remains non-vanishing and satisfies an equation similar to that for $C$ with $A_μ$ and $\bar{A}_μ$ 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, $C^0$, $C^μ$ and $C^{(μν)}$, are introduced by contracting the local-frame index of $C$ with those of the inverse vielbeins, $E_a^μ$ and $E_a^{(μν)}$, 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 $g_{μν}$ and the spin-3 field $ϕ_{μνλ}$ by means of the $ϕ$-expansion. The action integral for the pure spin-3 gravity in the metric-like formalism up to ${\cal O}(ϕ^2)$, obtained before in the literature, is re-derived. Then the matter action is re-expressed in terms of $g_{μν}$, $ϕ_{μνρ}$ and the covariant derivatives for spin-3 geometry. Spin-3 gauge transformation is extended to the matter fields.