Second-order formalism for 3D spin-3 gravity

A second-order formalism for the theory of 3D spin-3 gravity is considered. Such a formalism is obtained by solving the torsion-free condition for the spin connection omega a(a), and substituting the result into the action integral. In the first-order formalism of the spin-3 gravity defined in terms of SL(3, R) x SL(3, R) Chern-Simons (CS) theory, however, the generalized torsion-free condition cannot be easily solved for the spin connection, because the vielbein e(mu)(a) itself is not invertible. To circumvent this problem, extra vielbein-like fields e((mu nu))(alpha) are introduced as a functional of e(a). New sets of affine-like connections Gamma(n)(mu M) are defined in terms of the metric-like fields, and a generalization of the Riemann curvature tensor is also presented. In terms of this generalized Riemann tensor the action integral in the second-order formalism is expressed. The transformation rules of the metric and the spin-3 gauge field under the generalized diffeomorphims are obtained explicitly. As in Einstein gravity, the new affine-like connections are related to the spin connection by a certain gauge transformation and a gravitational CS term expressed in terms of the new connections is also presented.