Holographic Duality for 3D Spin-3 Gravity Coupled to Scalar Field

Abstract

The 3d spin-3 gravity theory is holographically dual to a 2d ${\mathcal{W}}_3$-extended CFT. In a large-c limit the symmetry algebra of the CFT reduces to $SU(1,2)\times SU(1,2)$. On the ground of symmetry the dual bulk space-time will be given by an 8d group manifold $SU(1,2)$. Hence we need to introduce five extra coordinates in addition to three ordinary ones. The 3d space-time is a 3d hyper-surface $\Sigma$ embedded at constant values of the extra variables. Operators in the CFT at the boundary of $\Sigma$ are expressed in terms of $\mathcal{W}$ descendants of the operators at the boundary of ${\Sigma }_0$, 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 $\mathcal{W}$-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 $su(1,2)$. 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. An 8d local frame is introduced and the equations of motion for the 8d connections $A_{\mu }$, ${\overline{A}}_{\mu }$ are solved. By restricting those solutions onto $\Sigma$, flat connections in 3d $SL(3,\mathbb{R})\times SL(3,\mathbb{R})$ Chern-Simons theory are obtained and new 3d black hole solutions with and without spin-3 charge are found by this method.

arXiv comment44 pages, 1 figure

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