As argued in [78], introducing a non-minimally coupled scalar field, three-dimensional Einstein gravity can be extended by infinite families of theories which admit simple analytic generalizations of the charged BTZ black hole. Depending on the gravitational couplings, the solutions may describe black holes with one or several horizons and with curvature or BTZ-like singularities. In other cases, the metric function behaves as $\textit{f}$ ($\textit{r}$) $$ \overset{\left(r\to 0\right)}{\sim } $$ $$ \mathcal{O}\left({r}^{2s}\right) $$ with $\textit{s}$ ≥ 1, and the black holes are completely regular — a feature unique to three dimensions. Regularity arises generically i.e., without requiring any fine-tuning of parameters. In this paper we show that all these theories satisfy Birkhoff theorems, so that the most general spherically-symmetric solutions are given by the corresponding static black holes. We perform a thorough characterization of the Penrose diagrams of the solutions, finding a rich structure which includes, in particular, cases which tessellate the plane and others in which the diagrams cannot be drawn in a single plane. We also study the motion of probe particles on the black holes, finding that observers falling to regular black holes reach the center after a finite proper time. Contrary to the singular cases, the particles are not torn apart by tidal forces, so they oscillate between antipodal points describing many-universe orbits. We argue that in those cases the region $\textit{r}$ = 0 can be interpreted as a horizon with vanishing surface gravity, giving rise to generic inner-extremal regular black hole solutions. We also analyze the deep interior region of the solutions identifying the presence of Kasner eons and the conditions under which they take place. Finally, we construct new black hole solutions in the case in which infinite towers of terms are included in the action.

Spain

Pablo Bueno

Ecuador

Oscar Andino

Chile

Javier Moreno

Guido Velde

On regular charged black holes in three dimensions

The vacuum entanglement entropy of a general conformal field theory (CFT) in $\textit{d}$ = 5 spacetime dimensions contains a universal term, $\textit{F}$($\textit{A}$), which has a complicated and non-local dependence on the geometric details of the region $\textit{A}$ and the theory. Analogously to the previously known $\textit{d}$ = 3 case, we prove that for CFTs in $\textit{d}$ = 5 which are holographically dual to Einstein gravity, $\textit{F}$($\textit{A}$) 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 $\textit{A}$ 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 $\textit{d}$ = 3 case, $\textit{F}$($\textit{A}$) is not globally minimized by a round ball $\textit{A}$ = 𝔹 . Rather, $\textit{F}$($\textit{A}$) can take arbitrarily positive and negative values as a function of $\textit{A}$. Hence, we conclude that the round ball is not a global minimizer of $\textit{F}$($\textit{A}$) for general five-dimensional CFTs.

Giorgos Anastasiou

Ignacio Araya

Rodrigo Olea

Belgium

Alejandro Lopez

Higher-dimensional Willmore energy as holographic entanglement entropy

We show via an explicit construction how an infinite tower of higher-curvature corrections generically leads to a resolution of the Schwarzschild singularity in any spacetime dimension <math altimg='si1.svg'><mi>D</mi><mo>&#8805;</mo><mn>5</mn></math>. The theories we consider have two key properties that ensure the results are general and robust: (1) they provide a basis for (vacuum) gravitational effective field theory in five and higher-dimensions, (2) for each value of the mass, they have a unique static spherically symmetric solution. We present several exact solutions of the theories that include the Hayward black hole and metrics similar to the Bardeen and Dymnikova ones. Unlike previous constructions, these regular black holes arise as vacuum solutions, as we include no matter fields whatsoever in our analysis. We show how the black hole thermodynamics can be studied in a completely universal and unambiguous way for all solutions.

Pablo A. Cano

Robie A. Hennigar

Regular black holes from pure gravity

The mutual information, $\textit{I}$ , of general spacetime regions is expected to capture the full data of any conformal field theory (CFT). For spherical regions, this data can be accessed from long-distance expansions of the mutual information of pairs of regions as well as of suitably chosen linear combinations of mutual informations involving more than two regions and their unions — namely, the $\textit{N}$-partite information, $\textit{I}$ . In particular, the leading term in the $\textit{I}$ long-distance expansion is fully determined by the spin and conformal dimension of the lowest-dimensional primary of the theory. When the operator is a scalar, an analogous formula for the tripartite information $\textit{I}$ contains information about the OPE coefficient controlling the fusion of such operator into its conformal family. When it is a fermionic field, the coefficient of the leading term in $\textit{I}$ vanishes instead. In this paper we present an explicit general formula for the long-distance four-partite information $\textit{I}$ of general CFTs whose lowest-dimensional operator is a fermion $\textit{ψ}$. The result involves a combination of four-point and two-point functions of $\textit{ψ}$ and $$ \overline{\psi} $$ evaluated at the locations of the regions. We perform explicit checks of the formula for a (2 + 1)-dimensional free fermion in the lattice finding perfect agreement. The generalization of our result to the $\textit{N}$-partite information (for arbitrary $\textit{N}$) is also discussed. Similarly to $\textit{I}$ , we argue that $\textit{I}$ vanishes identically at leading order for general fermionic theories, while the $\textit{I}$ with $\textit{N}$ = 7$\textit{,}$ 9$\textit{,}$ … only vanish when the theory is free.

Netherlands

César Agón

Long-distance $\textit{N}$-partite information for fermionic CFTs

In the vicinity of spacelike singularities, general relativity predicts that the metric behaves, at each point, as a Kasner space which undergoes a series of “Kasner epochs” and “eras” characterized by certain transition rules. The period during which this process takes place defines a “Kasner eon,” which comes to an end when higher-curvature or quantum effects become relevant. When higher-curvature densities are included in the action, spacetime can undergo transitions into additional Kasner eons. During each eon, the metric behaves locally as a Kasner solution to the higher-curvature density controlling the dynamics. In this paper we identify the presence of Kasner eons in the interior of static and spherically symmetric Lovelock gravity black holes. We determine the conditions under which eons occur and study the Kasner metrics which characterize them, as well as the transitions between them. We show that the null energy condition implies a monotonicity property for the effective Kasner exponent at the end of the Einsteinian eon. We also characterize the Kasner solutions of more general higher-curvature theories of gravity. In particular, we observe that the Einstein gravity condition that the sum of the Kasner exponents adds up to 1, <math><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>D</mi><mo>−</mo><mn>1</mn></mrow></msubsup><msub><mi>p</mi><mi>i</mi></msub><mo>=</mo><mn>1</mn></math>, admits a universal generalization in the form of a family of Kasner metrics satisfying <math><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>D</mi><mo>−</mo><mn>1</mn></mrow></msubsup><msub><mi>p</mi><mi>i</mi></msub><mo>=</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></math> which exists for any order-<math><mi>n</mi></math> higher-curvature density and in general dimensions.

Ming-Da Li

Kasner eons in Lovelock black holes

We present a top-down construction of a three-dimensional nonlocal theory of massive gravity. This “nonlocal massive gravity” (NLMG) is obtained as the gravitational theory induced by Einstein gravity on a brane inserted in anti–de Sitter space modified by an overall minus sign. The theory involves an infinite series of increasingly complicated higher-derivative corrections to the Einstein-Hilbert action, with the quadratic term coinciding with new massive gravity. We obtain an analytic formula for the quadratic action of NLMG and show that its linearized spectrum consists of an infinite tower of positive-energy massive spin-2 modes. We compute the Newtonian potential and show that the introduction of the infinite series of terms makes it behave as <math><mo>∼</mo><mn>1</mn><mo>/</mo><mi>r</mi></math> at short distances, as opposed to the logarithmic behavior encountered when the series is truncated at any finite order. We use this and input from brane-world holography to argue that the theory may contain asymptotically flat black hole solutions.

Nonlocal Massive Gravity from Einstein Gravity

We study various aspects of higher-curvature theories of gravity built from contractions of the metric, the Riemann tensor, and the covariant derivative, <math><mi>L</mi><mo>(</mo><msup><mi>g</mi><mrow><mi>a</mi><mi>b</mi></mrow></msup><mo>,</mo><msub><mi>R</mi><mrow><mi>a</mi><mi>b</mi><mi>c</mi><mi>d</mi></mrow></msub><mo>,</mo><msub><mo>∇</mo><mi>a</mi></msub><mo>)</mo></math>. We characterize the linearized spectrum of these theories and compute the modified Newton potential in the general case. Then, we present the first examples of generalized quasitopological (GQT) gravities involving covariant derivatives of the Riemann tensor. We argue that they always have second-order equations on maximally symmetric backgrounds. Focusing on four spacetime dimensions, we find new densities of that type involving eight and ten derivatives of the metric. In the latter case, we find new modifications of the Schwarzschild black hole. These display thermodynamic properties which depart from the ones of polynomial GQT black holes. In particular, the relation between the temperature and the mass of small black holes, <math><mi>T</mi><mo>∼</mo><msup><mi>M</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></math>, which universally holds for general polynomial GQT modifications of Einstein gravity, gets modified in the presence of the new density with covariant derivatives to <math><mi>T</mi><mo>∼</mo><msup><mi>M</mi><mn>3</mn></msup></math>. Finally, we consider brane-world gravities induced by Einstein gravity in the AdS bulk. We show that the effective quadratic action for the brane-world theory involving arbitrary high-order terms in the action can be written explicitly in a closed form in terms of Bessel functions. We use this result to compute the propagator of metric perturbations on the brane and its pole structure in various dimensions, always finding infinite towers of ghost modes, as well as tachyons and more exotic modes in some cases.

Sergio E. Aguilar-Gutierrez

Quim Llorens

Aspects of higher-curvature gravities with covariant derivatives

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

Oscar Lasso Andino

Israel

Conformal Bounds in Three Dimensions from Entanglement Entropy

We construct generalized symmetries for linearized Einstein gravity in arbitrary dimensions. First-principle considerations in quantum field theory force generalized symmetries to appear in dual pairs. Verifying this prediction helps us find the full set of nontrivial conserved charges—associated, in equal parts, with 2-form and (<math><mrow><mi>D</mi><mo>−</mo><mn>2</mn></mrow></math>)-form currents. Their total number is <math><mrow><mi>D</mi><mo>(</mo><mi>D</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></math>. We compute the quantum commutators of pairs of dual charges, showing that they are nonvanishing for regions whose boundaries are nontrivially linked with each other and zero otherwise, as expected on general grounds. We also consider general linearized higher-curvature gravities. These propagate, in addition to the usual graviton, a spin-0 mode as well as a massive ghostlike spin-2 mode. When the latter is absent, the theory is unitary and the dual-pairs principle is respected. In particular, we find that the number and types of charges remain the same as for Einstein gravity, and that they correspond to continuous generalizations of the Einsteinian ones.

Valentin Benedetti

Javier M. Magan

Generalized Symmetries for Generalized Gravitons

We present several new results for the N-partite information, I  N , of spatial regions in the ground state of d-dimensional conformal field theories. First, we show that I  N  can be written in terms of a single N-point function of twist operators. Using this, we argue that in the limit in which all mutual separations are much greater than the regions sizes, the N-partite information scales as I  N  ~ r  −2N∆, where r is the typical distance between pairs of regions and ∆ is the lowest primary scaling dimension. In the case of spherical entangling surfaces, we obtain a completely explicit formula for the I 4 in terms of 2-, 3- and 4-point functions of the lowest-dimensional primary. Then, we consider a three- dimensional scalar field in the lattice. We verify the predicted long-distance scaling and provide strong evidence that I  N  is always positive for general regions and arbitrary N for that theory. For the I 4, we find excellent numerical agreement between our general formula and the lattice result for disk regions. We also perform lattice calculations of the mutual information for more general regions and general separations both for a free scalar and a free fermion, and conjecture that, normalized by the corresponding disk entanglement entropy coefficients, the scalar result is always greater than the fermion one. Finally, we verify explicitly the equality between the N-partite information of bulk and boundary fields in holographic theories for spherical entangling surfaces in general dimensions.