The vacuum entanglement entropy of a general conformal field theory (CFT) in = 5 spacetime dimensions contains a universal term, ( ), which has a complicated and non-local dependence on the geometric details of the region and the theory. Analogously to the previously known = 3 case, we prove that for CFTs in = 5 which are holographically dual to Einstein gravity, ( ) is equal to a four-dimensional version of the “Willmore energy” associated to a doubled and closed version of the Ryu-Takayanagi (RT) surface of embedded in ℝ . This generalized Willmore energy is shown to arise from a conformal-invariant codimension-two functional obtained by evaluating six-dimensional Conformal Gravity on the conically-singular orbifold of the replica trick. The new functional involves an integral over the doubled RT surface of a linear combination of three quartic terms in extrinsic curvatures and is free from ultraviolet divergences by construction. We verify explicitly the validity of our new formula for various entangling regions and argue that, as opposed to the = 3 case, ( ) is not globally minimized by a round ball = 𝔹 . Rather, ( ) can take arbitrarily positive and negative values as a function of . Hence, we conclude that the round ball is not a global minimizer of ( ) for general five-dimensional CFTs.

Chile

Giorgos Anastasiou

Ignacio Araya

Spain

Pablo Bueno

Javier Moreno

Rodrigo Olea

Belgium

Alejandro Lopez

Higher-dimensional Willmore energy as holographic entanglement entropy

The entanglement entropy of an arbitrary spacetime region <math><mi>A</mi></math> in a three-dimensional conformal field theory (CFT) contains a constant universal coefficient, <math><mi>F</mi><mo>(</mo><mi>A</mi><mo>)</mo></math>. For general theories, the value of <math><mi>F</mi><mo>(</mo><mi>A</mi><mo>)</mo></math> is minimized when <math><mi>A</mi></math> is a round disk, <math><msub><mi>F</mi><mn>0</mn></msub></math>, and in that case it coincides with the Euclidean free energy on the sphere. We conjecture that, for general CFTs, the quantity <math><mi>F</mi><mo>(</mo><mi>A</mi><mo>)</mo><mo>/</mo><msub><mi>F</mi><mn>0</mn></msub></math> is bounded above by the free scalar field result and below by the Maxwell field one. We provide strong evidence in favor of this claim and argue that an analogous conjecture in the four-dimensional case is equivalent to the Hofman-Maldacena bounds. In three dimensions, our conjecture gives rise to similar bounds on the quotients of various constants characterizing the CFT. In particular, it implies that the quotient of the stress-tensor two-point function coefficient and the sphere free energy satisfies <math><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>/</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>≤</mo><mn>3</mn><mo>/</mo><mo>(</mo><mn>4</mn><msup><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>log</mi><mn>2</mn><mo>−</mo><mn>6</mn><mi>ζ</mi><mo>[</mo><mn>3</mn><mo>]</mo><mo>)</mo><mo>≃</mo><mn>0.14887</mn></mrow></math> for general CFTs. We verify the validity of this bound for free scalars and fermions, general <math><mi>O</mi><mo>(</mo><mi>N</mi><mo>)</mo></math> and Gross-Neveu models, holographic theories, <math><mi>N</mi><mo>=</mo><mn>2</mn></math> Wess-Zumino models and general ABJM theories.

Argentina

Horacio Casini

Ecuador

Oscar Lasso Andino

Israel

Conformal Bounds in Three Dimensions from Entanglement Entropy

The entanglement entropy corresponding to a smooth region in general three-dimensional CFTs contains a constant universal term, −F ⊂ S EE. For a disk region, F|disk ≡ F 0 coincides with the free energy on 𝕊3 and provides an RG-monotone for general theories. As opposed to the analogous quantity in four dimensions, the value of F generally depends in a complicated (and non-local) way on the geometry of the region and the theory under consideration. For small geometric deformations of the disk in general CFTs as well as for arbitrary regions in holographic theories, it has been argued that F is precisely minimized by disks. Here, we argue that F is globally minimized by disks with respect to arbitrary regions and for general theories. The proof makes use of the strong subadditivity of entanglement entropy and the geometric fact that one can always place an osculating circle within a given smooth entangling region. For topologically non-trivial entangling regions with n  B  boundaries, the general bound can be improved to F ≥ n  B  F 0. In addition, we provide accurate approximations to F valid for general CFTs in the case of elliptic regions for arbitrary values of the eccentricity which we check against lattice calculations for free fields. We also evaluate F numerically for more general shapes in the so-called “Extensive Mutual Information model”, verifying the general bound.

Oscar Andino

United States

Disks globally maximize the entanglement entropy in 2 + 1 dimensions

We derive a covariant expression for the renormalized holographic entanglement entropy for conformal field theories (CFTs) dual to quadratic curvature gravity in arbitrary dimensions. This expression is written as the sum of the bare entanglement entropy functional obtained using standard conical defect techniques, and a counterterm defined at the boundary of the extremal surface of the functional. The latter corresponds to the cod-2 self-replicating part of the extrinsic counterterms when evaluated on the replica orbifold. This renormalization method isolates the universal terms of the holographic entanglement entropy functional. We use it to compute the standard <math><mi>C</mi></math>-function candidate for CFTs of arbitrary dimension, and the type-B anomaly coefficient <math><mi>c</mi></math> for four-dimensional CFTs.

Ignacio J. Araya

France

David Rivera-Betancour

Renormalized holographic entanglement entropy for quadratic curvature gravity

We study the holographic entanglement entropy of deformed entangling regions in three-dimensional CFTs dual to Einstein-AdS gravity, using a renormalization scheme based on the addition of extrinsic counterterms. In this prescription, when even- dimensional manifolds are considered, the universal contribution to the entanglement entropy is identified as the renormalized volume of the Ryu-Takayanagi hypersurface, which is written as the sum of a topological and a curvature term. It is shown that the change in the renormalized entanglement entropy due to the deformation of the entangling surface is encoded purely in the curvature contribution. In turn, as the topological part is given by the Euler characteristic of the Ryu-Takayanagi surface, it remains shape independent. Exploiting the covariant character of the extrinsic counterterms, we apply the renormalization scheme for the case of deformed entangling regions in AdS4/CFT3, recovering the results found in the literature. Finally, we provide a derivation of the relation between renormalized entanglement entropy and Willmore energy. The presence of a lower bound of the latter makes manifest the relation between the AdS curvature of the Ryu-Takayanagi surface and the strong subadditivity property.

Shape dependence of renormalized holographic entanglement entropy

Generalized quasi-topological gravities (GQTGs) are higher-curvature extensions of Einstein gravity characterized by the existence of non-hairy generalizations of the Schwarzschild black hole which satisfy g  tt  g  rr  = –1, as well as for having second-order linearized equations around maximally symmetric backgrounds. In this paper we provide strong evidence that any gravitational effective action involving higher-curvature corrections is equivalent, via metric redefinitions, to some GQTG. In the case of theories involving invariants constructed from contractions of the Riemann tensor and the metric, we show this claim to be true as long as (at least) one non-trivial GQTG invariant exists at each order in curvature-and extremely conclusive evidence suggests this is the case in general dimensions. When covariant derivatives of the Riemann tensor are included, the evidence provided is not as definitive, but we still prove the claim explicitly for all theories including up to eight derivatives of the metric as well as for terms involving arbitrary contractions of two covariant derivatives of the Riemann tensor and any number of Riemann tensors. Our results suggest that the physics of generic higher-curvature gravity black holes is captured by their GQTG counterparts, dramatically easier to characterize and universal. As an example, we map the gravity sector of the Type-IIB string theory effective action in AdS5 at order 𝒪 (α′3) to a GQTG and show that the thermodynamic properties of black holes in both frames match.

Pablo Cano

Ángel Murcia

All higher-curvature gravities as Generalized quasi-topological gravities