Second-Order Formalism for 3D Spin-3 Gravity

Understand

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_{\mu}, 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^a_{\mu} itself is not invertible.
• To circumvent this problem, extra vielbein-like fields e^a_{\mu\nu} are introduced as a functional of e^a_{\mu}.