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

Understand

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.