We derive the universal terms of entanglement entropy for 6d CFTs by applying the holographic and the field theoretical approaches, respectively. Our formulas are conformal invariant and agree with the results of [37, 38]. Remarkably, we find that the holographic and the field theoretical results match exactly for the C 2 and Ck 2 terms, where C and k denote the Weyl tensor and the extrinsic curvature, respectively. As for the k 4 terms, we meet the splitting problem of the conical metrics. The splitting problem in the bulk can be fixed by equations of motion. As for the splitting on the boundary, we assume the general forms and find that there indeed exists suitable splitting which can make the holographic and the field theoretical k 4 terms match. Since we have much more equations than the free parameters, the match for k 4 terms is non-trivial.
We derive the universal terms of entanglement entropy for 6d CFTs by applying the holographic and the field theoretical approaches, respectively. Our formulas are conformal invariant and agree with the results of [37, 38]. Remarkably, we find that the holographic and the field theoretical results match exactly for the C 2 and Ck 2 terms, where C and k denote the Weyl tensor and the extrinsic curvature, respectively. As for the k 4 terms, we meet the splitting problem of the conical metrics. The splitting problem in the bulk can be fixed by equations of motion. As for the splitting on the boundary, we assume the general forms and find that there indeed exists suitable splitting which can make the holographic and the field theoretical k 4 terms match. Since we have much more equations than the free parameters, the match for k 4 terms is non-trivial.
We study holographic entanglement entropy in 3-dimensional AdS gravity with conformal boundary conditions which fix the conformal class of the boundary metric and its extrinsic curvature, K, while leaving the Weyl mode dynamical. Extending Lewkowycz-Maldacena-Dong's replica construction, we derive the corresponding holographic entanglement entropy formula. We show that the fluctuating Weyl mode does not contribute additional entropy. The entropy of the full boundary is therefore the Bekenstein-Hawking entropy, and the entropy of a boundary subregion continues to obey the Ryu-Takayanagi prescription, namely, the area of a minimal surface divided by 4GN. We carry out explicit calculations for global AdS, rotating and non-rotating BTZ geometries. For an interval in AdS3 we find that the K-dependent holographic entanglement entropy is governed by cm=2GN3ℓ. For a thermal state at high conformal temperatures we find that, the entropy is governed by ceff=2GN3ℓ2Kℓ−K2ℓ2−4, the same effective central charge that governs the Cardy-like density of states, in agreement with previous results in the literature. Finally, we also compute the entanglement entropy directly from the conjectured dual boundary theory - a holographic CFT coupled with time-like Liouville theory and deformed by a marginal TTˉ like operator - and find SEE=3ceffln(ε2Rsinϕ0). This result for the state with no operator insertions (vacuum state ), provides an independent boundary realization of ceff while clarifying that this state is not the state dual to global AdS.
Entanglement entropy in holographic conformal field theories with boundaries of generic spacetime signature cannot be described entirely within the standard holographic duality to a real bulk spacetime. We show, both in static and time-dependent setups, that an exact match with the conformal field theory predictions in two boundary dimensions entails an extension of the dual spacetime to complex coordinates. As a consequence, relevant bulk objects such as Ryu-Takayanagi surfaces and branes are in general complex, yet anchored on a real asymptotic boundary preserving the physical meaning of field-theoretical quantities. We also provide a third numerical check of the holographic prediction with a two-dimensional system of free Gaussian fermions exhibiting the same asymptotic regimes across different entanglement phases.
A bstract We study the structure of divergences and universal terms of the entanglement and Rényi entropies for singular regions. First, we show that for (3 + 1)-dimensional free conformal field theories (CFTs), entangling regions emanating from vertices give rise to a universal contribution Snuniv=−8π1fb(n)∫γk2log2(R/δ) S n univ = − 1 8 π f b n ∫ γ k 2 log 2 R / δ , where γ is the curve formed by the intersection of the entangling surface with a unit sphere centered at the vertex, and k the trace of its extrinsic curvature. While for circular and elliptic cones this term reproduces the general-CFT result, it vanishes for polyhedral corners. For those, we argue that the universal contribution, which is logarithmic, is not controlled by a local integral, but rather it depends on details of the CFT in a complicated way. We also study the angle dependence for the entanglement entropy of wedge singularities in 3+1 dimensions. This is done for general CFTs in the smooth limit, and using free and holographic CFTs at generic angles. In the latter case, we show that the wedge contribution is not proportional to the entanglement entropy of a corner region in the (2 + 1)-dimensional holographic CFT. Finally, we show that the mutual information of two regions that touch at a point is not necessarily divergent, as long as the contact is through a sufficiently sharp corner. Similarly, we provide examples of singular entangling regions which do not modify the structure of divergences of the entanglement entropy compared with smooth surfaces.
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.
We study universal features in the shape dependence of entanglement entropy in the vacuum state of a conformal field theory (CFT) on R1,d−1 . We consider the entanglement entropy across a deformed planar or spherical entangling surface in terms of a perturbative expansion in the infinitesimal shape deformation. In particular, we focus on the second order term in this expansion, known as the entanglement density. This quantity is known to be non-positive by the strong-subadditivity property. We show from a purely field theory calculation that the non-local part of the entanglement density in any CFT is universal, and proportional to the coefficient C T appearing in the two-point function of stress tensors in that CFT. As applications of our result, we prove the conjectured universality of the corner term coefficient CTσ=24π2 in d = 3 CFTs, and the holographic Mezei formula for entanglement entropy across deformed spheres.
The entanglement entropy of a generic d-dimensional conformal field theory receives a regulator independent contribution when the entangling surface contains a (hyper)conical singularity of opening angle Ω, codified in a function a (d)(Ω). In arXiv:1505.04804 , we proposed that for three-dimensional conformal field theories, the coefficient σ (3) characterizing the limit where the surface becomes smooth is proportional to the central charge C T appearing in the two-point function of the stress tensor. In this paper, we prove this relation for general three-dimensional holographic theories, and extend the result to general dimensions. In particular, we define a generalized coefficient σ (d) to characterize the almost smooth limit of a (hyper)conical singularity in entangling surfaces in higher dimensions. We show then that this coefficient is universally related to C T for general holographic theories and provide a general formula for the ratio σ (d) /C T in arbitrary dimensions. We conjecture that the latter ratio is universal for general CFTs. Further, based on our recent results in arXiv:1507.06997 , we propose an extension of this relation to general Rényi entropies, which we show passes several consistency checks in d = 4 and 6.