Geometric aspect of three-dimensional Chern–Simons-like gravity

Abstract

In this paper, we detailed study the geometric aspect of Chern–Simons-like gravity in three dimensions. The vacua are provided by three dimensional conformally flat manifold, which admit a special configuration, a two dimensional system (M2,h,Φ) consisting of metric h and scalar field Φ, by dimensional reduction. For this system we define the quasi-local mass. An interesting observation is that this system contains certain two dimensional dilaton gravity at the classical level. Via AdS/CFT, we check the Weyl anomaly and diffeomorphism anomaly for the boundary theory. We study the linearization of Chern–Simons-like gravity around the background with constant scalar curvature via linear Cotton tensor. If the background manifold is of positive constant scalar curvature, we show that there is no solutions for linearized vacuum equation. To find the solutions of the linear gravity with respect to the background with negative constant scalar curvature, we need to solve some Schrödinger-type equations. Related to supersymmetric quantum mechanics, one can find some exact solutions and some dualities between different components or modes of solutions. And it is also can be related to Seiberg–Witten theory via Picard–Fuchs equation in terms of WKB approximation. We discuss the ADM-type charge under the context of Chern–Simons-like gravity. Finally, we extend the Chern–Simons-like gravity to the supermanifolds by embedding the structure group of three-manifold into the body of othosymplectic supergroup.