Generalized gravitational entropy without replica symmetry

Abstract: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.

Section: Jhep10(2015)049mentioning

confidence: 94%

“…Our metric eq. (2.13) can be regarded as a special case of [44] with Z n symmetry. Inspired by [7], it is expected that the splitting problem can be fixed by equations of motion.…”

Section: Jhep10(2015)049mentioning

confidence: 99%

Universal terms of entanglement entropy for 6d CFTs

Miao

2015

“…The advantage of this trick is that the on-shell action will be given simply by evaluating these contributions. For Einstein-Hilbert gravitational dynamics we evaluate the Gibbons-Hawking contribution from e q ( ) 21 20 We should note that such terms are sometimes desired in order to cancel subleading divergences in higher derivatives theories [26]. 21 To prevent notational clutter, we drop the integration measure for simplicity in the formulae henceforth.…”

Section: Jhep11(2016)028mentioning

confidence: 99%

“…On the Cauchy slice as we reverse the evolution, we have to provide appropriate matching (boundary) conditions. 26 As described in section 2, we may move Σ t itself as long as J − [∂A] is unmodified. Let us say, for definiteness, we make a particular choice and stick with it w.l.o.g.…”

Section: Covariant Generalization To Time-dependent Situationsmentioning

confidence: 99%

“…25 Similar statements apply to states prepared at t = t0 < 0, for instance by a Euclidean path integral, and then evolved up to t = 0 perhaps with sources, etc. 26 These boundary conditions guarantee that we have a well defined variational principle and that the variation of the action doesn't contain boundary terms atΣt. however does not single out a bulk Cauchy slice, since the boundary time coordinate does not extend uniquely into the bulk.…”

Section: Covariant Generalization To Time-dependent Situationsmentioning

confidence: 99%

See 1 more Smart Citation

Deriving covariant holographic entanglement

Dong

Lewkowycz

Rangamani

2016

265

378

We provide a gravitational argument in favour of the covariant holographic entanglement entropy proposal. In general time-dependent states, the proposal asserts that the entanglement entropy of a region in the boundary field theory is given by a quarter of the area of a bulk extremal surface in Planck units. The main element of our discussion is an implementation of an appropriate Schwinger-Keldysh contour to obtain the reduced density matrix (and its powers) of a given region, as is relevant for the replica construction. We map this contour into the bulk gravitational theory, and argue that the saddle point solutions of these replica geometries lead to a consistent prescription for computing the field theory Rényi entropies. In the limiting case where the replica index is taken to unity, a local analysis suffices to show that these saddles lead to the extremal surfaces of interest. We also comment on various properties of holographic entanglement that follow from this construction.

Entropy, extremality, euclidean variations, and the equations of motion

Dong

Lewkowycz

2018

175

260

Abstract:We study the Euclidean gravitational path integral computing the Rényi entropy and analyze its behavior under small variations. We argue that, in Einstein gravity, the extremality condition can be understood from the variational principle at the level of the action, without having to solve explicitly the equations of motion. This set-up is then generalized to arbitrary theories of gravity, where we show that the respective entanglement entropy functional needs to be extremized. We also extend this result to all orders in Newton's constant G N , providing a derivation of quantum extremality. Understanding quantum extremality for mixtures of states provides a generalization of the dual of the boundary modular Hamiltonian which is given by the bulk modular Hamiltonian plus the area operator, evaluated on the so-called modular extremal surface. This gives a bulk prescription for computing the relative entropies to all orders in G N . We also comment on how these ideas can be used to derive an integrated version of the equations of motion, linearized around arbitrary states.