Metric-like formalism for matter fields coupled to 3D higher spin gravity

Abstract

The action integral for a matter system composed of 0- and 2-forms, C and Bμν, topologically coupled to 3D spin-3 gravity is considered first in the frame-like formalism. The field C satisfies an equation of motion, , where Aμ and are the Chern–Simons gauge fields. With a suitable gauge fixing of a new local symmetry and diffeomorphism, only one component of Bμν, say Bϕr, remains non-vanishing and satisfies . These equations are the same as those for 3D (free) Vasiliev scalars, C and . The spin connection is eliminated by solving the equation of motion for the total action, and 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, C0, Cμ and C(μν), are introduced by contracting the local-frame index of C with those of the inverse vielbeins, and , which were defined by the present authors in (2013 Class. Quantum Grav. 30 035003). 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 gμν and the spin-3 field ϕμνλ by means of the ϕ-expansion. The action integral for the pure spin-3 gravity in the metric-like formalism up to , obtained before in the literature, is re-derived. Then the matter action is re-expressed in terms of gμν, ϕμνρ and the covariant derivatives for spin-3 geometry. Spin-3 gauge transformation is extended to the matter fields. It is also found that the action of the matter-coupled theory in the metric-like formalism has larger symmetry than that of the pure spin-3 gravity. The matter-coupled theory in the metric-like formalism is invariant under the ordinary diffeomorphisms, because the vielbein and the spin connection are covariant vectors of diffeomorphisms. They are also gauge fields for the local translation. In the pure spin-3 gravity this symmetry provides the ordinary diffeomorphism and the spin-3 transformation in the metric-like formalism. When the matter is coupled, the local translation yields a new symmetry in the metric-like formalism, which does not contain diffeomorphism.