Abstract
Entanglement entropy of holographic CFTs is expected to play a crucial role in the reconstruction of semiclassical bulk gravity. We consider the entanglement entropy of spherical regions of vacuum, which is known to contain universal contributions. After perturbing the CFT with a relevant scalar operator, also the first order change of this quantity gives a universal term which only depends on a discrete set of basic CFT parameters. We show that in gravity this statement corresponds to the uniqueness of the ghost-free graviton propagator on an AdS background geometry. While the gravitational dynamics in this context contains little information about the structure of the bulk theory, there is a discrete set of dimensionless parameters of the theory which determines the entanglement entropy. We argue that for every (not necessarily holographic) CFT, any reasonable gravity model can be used to compute this particular entanglement entropy. We elucidate how this statement is consistent with AdS/CFT and also give various generalizations. On the one hand this illustrates the remarkable usefulness of geometric concepts for understanding entanglement in general CFTs. On the other hand, it provides hints as to what entanglement data can be expected to provide enough information to distinguish, e.g., bulk theories with different higher curvature couplings.
Highlights
One hint is the fact that a local bulk metric should be represented in the boundary CFT in a non-local fashion
Even for non-holographic CFTs there is evidence that geometric concepts sometimes provide the most natural and efficient way of computing entanglement entropy. The first such statement is the realization that vacuum entanglement entropy of spherical regions can be conformally mapped to thermal entropy which sometimes has a natural interpretation in terms of black hole thermodynamics [10]
If spherical region entanglement entropy in deformations of any arbitrary CFT can be computed using linearized Einstein gravity [11], it follows that this quantity is certainly not sufficient to distinguish even CFTs with a semiclassical gravity dual from those without; let alone distinguish a bulk theory governed by Einstein gravity from any other higher derivative theory with the same spectrum of low energy excitations
Summary
We analyze universal features both of CFTs and of gravitational theories, which explain universal properties of entanglement entropy of ball shaped spatial regions in CFTs deformed by a scalar operator. These parameters can again be absorbed in a renormalization of dimensionless quantities such that the answer is indistinguishable from what one would have obtained in Einstein gravity minimally coupled to a scalar field This is the main universality statement studied : For holographically computing entanglement entropy of spherical regions in CFTs deformed by a relevant scalar operator, any physically acceptable theory of a spin-2 and a spin-0 field with appropriately tuned values of the couplings gives the same result as cosmological Einstein gravity minimally coupled to a scalar. This quantity does not distinguish between different higher curvature theories of gravity. In the rest of this section, we will explain these statements from the CFT and from the gravity side
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