Hořava-Lifshitz Holography

We derive the detailed balance condition as a solution to the Hamilton-Jacobi equation in the Hořava-Lifshitz gravity. This result leads us to propose the existence of the d-dimensional quantum field theory on the future boundary of the (d + 1)dimensional Hořava-Lifshitz gravity from the viewpoint of the holographic renormalization group. We also obtain a Ricci flow equation of the boundary theory as the holographic RG flow, which is the Hamilton equation in the bulk gravity, by tuning parameters in the theory. A renormalizable theory of gravity, so called the Hořava-Lifshitz gravity, has been attracting great interest since the advent of the seminal works of Hořava [1, 2]. This theory does not have the full diffeomorphism invariance, but the following anisotropic scaling with dynamical critical exponent z larger than the spatial dimensions d exists x → bx (i = 1, . . . , d) , t → bt . (1) This theory has a remarkable property such that it describes the non-relativistic theory of gravity in the UV regime, but becomes the Einstein gravity in the IR region. It was extensively applied to a resolution of the cosmological problem including inflation and nonGaussianity in [3–10], and new solutions were constructed in [11–15]. For the other recent progress, refer the reader to [16–27]. It becomes convenient in the following discussion to take the Wick rotation to the Euclidean space. The ADM decomposition of the (d+ 1)-dimensional Euclidean space is ds = Ndt + gij(dx i +N dt)(dx +N dt) , (2) and the extrinsic curvature for the spacelike slice is given by Kij = 1 N (ġij −∇iNj −∇jNi) . (3) The action of the Hořava-Lifshitz gravity is defined as [1, 2]