Abstract
We extend Maldacena’s argument, namely, obtaining Einstein gravity from Conformal Gravity, to six dimensional manifolds. The proof relies on a particular combination of conformal (and topological) invariants, which makes manifest the fact that 6D Conformal Gravity admits an Einstein sector. Then, by taking generalized Neumann boundary conditions, the Conformal Gravity action reduces to the renormalized Einstein-AdS action. These restrictions are implied by the vanishing of the traceless Ricci tensor, which is the defining property of any Einstein spacetime. The equivalence between Conformal and Einstein gravity renders trivial the Einstein solutions of 6D Critical Gravity at the bicritical point.
Highlights
Boundary data g(0)ij appear in the asymptotic expansion of the metric, by solving iteratively the Einstein field equations [8]
It is an appealing idea to try to link it to the existence of conformal structures defined throughout the manifold. This reasoning suggests that the renormalization of the volume/action might be encoded in a mathematical object –defined for Einstein spaces– but that can be embedded in a conformally invariant theory of gravity, i.e., a particular version of Conformal Gravity (CG)
We have shown that the LPP version of Conformal Gravity in 6D is classically equivalent to the Einstein-AdS action for Einstein spaces
Summary
Conformal Gravity is a theory invariant under local Weyl rescalings of the metric gμν → Ω2 (x) gμν and it is defined by a linear combination of the conformal invariants of the corresponding dimension. There is a unique choice of parameters where a Scwarzschild-AdS black hole is admitted in the solutions space of the theory, which reads c1 = 4c2 = −12c3 [40] This choice in 6D is denoted as Lu, Pang and Pope (LPP) CG, and the fact that it has an Einstein sector despite being a higher derivative theory can be understood based on the corresponding Lagrangian having no explicit dependence on Riemann or Riemann terms (up to the topological term in 6D). It has been shown that when an AAdS manifold with a conformally flat boundary is considered, the counterterms needed for the regularization of its volume acquire a closed form expression that depends explicitly on the extrinsic curvature of the boundary [43] This resummed form of the counterterms in even-D [44], is equivalent to the Euler term of the corresponding dimension, up to the Euler characteristic. Reducing the metric of a CG solution to the one of the Einstein subclass will allow us to constrain the conformal invariants to their restricted form, making the matching apparent
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