AdS₃ Action in the Dreibein Formalism

Abstract

We examine how to rewrite the (2 + 1)-dimensional AdS3 action with gravitational field into action with dreibein. We obtained the non-trivial Einstein equation with dreibein. While Brown-Henneaux rewrite the AdS3 action by using i) torsionless condition, ii) Einstein equation with dreibein. But we cannot use torsionless condition because the torsionless condition is automatically satisfied by imposing the dreibein hypothesis. Furthermore, we cannot use the Einstein equation with the dreibein to rewrite the AdS3 action. If the AdS3 action becomes in the Chern-Simons form, the equation of motion becomes trivial and there is no equation of motion, which contradicts with the non-trivial Einstein equation with dreibein. Therefore, the Brown-Henneaux’s procedure is not allowed, and we cannot obtain the Chern-Simons type action.