Holographic duality for 3d spin-3 gravity coupled to scalar field

Abstract The 3d spin-3 gravity theory is holographically dual to a 2d W 3 -extended CFT. In a large-c limit the symmetry algebra of the CFT reduces to S U ( 1 , 2 ) × S U ( 1 , 2 ) . On ground of symmetry the dual bulk space–time will be given by an 8d group manifold S U ( 1 , 2 ) . Hence we need to introduce five extra coordinates in addition to three ordinary ones. The 3d space–time is a 3d hyper-surface Σ embedded at constant values of the extra variables. Operators in the CFT at the boundary of Σ are expressed in terms of W descendants of the operators at the boundary of Σ 0 , where the extra variables vanish. In this paper it is shown that AdS/CFT correspondence for a scalar field coupled to 3d spin-3 gravity is realized in this auxiliary 8d space. A bulk-to-boundary propagator of a scalar field is found and a generating functional of boundary two-point functions of scalar W -descendant operators is obtained by using the classical action for the scalar field. Classically, the scalar field must satisfy both Klein–Gordon equation and a third-order differential equation, which are related to the quadratic and cubic Casimir operators of s u ( 1 , 2 ) . It is found that the coefficient function of the derivatives of the scalar field in the latter equation is the spin-3 gauge field, when restricted to the hypersurface. An action integral in the 8d auxiliary space for the 3d spin-3 gravity coupled to a scalar field is presented. In general, this 8d auxiliary space is a deformation of the manifold S U ( 1 , 2 ) . An 8d local frame is introduced and the equations of motion for the 8d connections A μ , A ‾ μ are solved. By restricting those solutions onto Σ, flat connections in 3d S L ( 3 , R ) × S L ( 3 , R ) Chern–Simons theory are obtained and new 3d black hole solutions with and without spin-3 charge are found by this method.