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

DOI10.1016/J.NUCLPHYSB.2019.02.005zbMATH Open1409.81089arXiv1902.02541OpenAlexW2911757634WikidataQ125574089 ScholiaQ125574089MaRDI QIDQ1731691

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Publication date: 13 March 2019

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Abstract: The 3d spin-3 gravity theory is holographically dual to a 2d ${calW}_3$-extended CFT. In a large-c limit the symmetry algebra of the CFT reduces to $SU(1,2)imesSU(1,2)$. 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 $calW$ descendants of the operators at the boundary of $Sigm{a}_0$, 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 $calW$-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 $SL(3,mathbbR)imesSL(3,mathbbR)$ Chern-Simons theory are obtained and new 3d black hole solutions with and without spin-3 charge are found by this method.

Full work available at URL: https://arxiv.org/abs/1902.02541

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