We construct a massive spin-2 theory in 2+1 dimensions that is immune to the bulk-boundary unitarity conflict in anti-de Sitter space and hence amenable to holography. The theory is an extension of Topologically Massive Gravity, just like the recently found Minimal Massive Gravity (MMG), but it has two massive helicity modes instead of a single one. The theory admits all the solutions of TMG with a redefined topological parameter. We calculate the Shapiro time-delay and show that flat-space (local) causality is not violated. We show that there is an interesting relation between the theory we present here (which we call MMG$_2$), MMG and the earlier New Massive Gravity (NMG): Namely, field equations of these theories are non-trivially related. We study the bulk excitations and boundary charges of the conformal field theory that could be dual to gravity. We also find the chiral gravity limit for which one of the massive modes becomes massless. The virtue of the model is that one does not have to go to the chiral limit to achieve unitarity in the bulk and on the boundary and the log-terms that appear in the chiral limit and cause instability do not exist in the generic theory.
Recently, an extension of the topologically massive gravity (TMG) in $2+1$ dimensions, dubbed as minimal massive gravity (MMG), was found which is free of the bulk-boundary unitarity clash that inflicts the former theory and all the other known three dimensional theories. Field equations of MMG differ from those of TMG at quadratic terms in the curvature that do not come from the variation of an action depending on the metric alone. Here we show that MMG is a unique theory and there does not exist a deformation of TMG or MMG at the cubic and quartic order (and beyond) in the curvature that is …
We look at solutions of Minimal Massive Gravity (MMG), a generalisation of Topologically Massive Gravity (TMG) that improves upon its holographic properties. It is shown that generically (in MMG parameter space) all conformally flat solutions of vacuum MMG are locally isometric to one of the two (A)dS vacua of the theory. We then couple a scalar field, and find that domain wall solutions can only interpolate between these two vacua precisely when the bulk graviton is tachyonic. Finally, we find a non-BTZ AdS black hole solution satisfying Brown-Henneaux boundary conditions, which lies within t…
We study the complexity growth by using complexity = action (CA) proposal in Minimal Massive 3D Gravity(MMG) model which is proposed for resolving the bulk-boundary clash problem of Topologically Massive Gravity(TMG). We observe that the rate of the complexity growth for BTZ black hole saturates the proposed bound by physical mass of the BTZ black hole in the MMG model, when the angular momentum parameter and the inner horizon of black hole goes to zero.
A geometrical analysis of the bulk and anti-de Sitter boundary unitarity conditions of 3D "Minimal Massive Gravity" (MMG) (which evades the "bulk/boundary clash" of Topologically Massive Gravity) is used to extend and simplify previous results, showing that unitarity selects, up to equivalence, a connected region in parameter space. We also initiate the study of flat-space holography for MMG. Its relevant flat space limit is a deformation of 3D conformal gravity; the deformation is both non-linear and non-conformal, implying a linearisation instability.
We find the excitations and construct the conserved charges ( mass and angular momentum) of the recently found Minimal Massive Gravity (MMG) in 2+1 dimensions in asymptotically Anti-de Sitter (AdS) spacetimes. The field equation of the theory does not come from an action and hence lacks the required Bianchi Identity needed to define conserved charges. But the theory, which also provides a healthy extension of the Topologically Massive Gravity in the bulk and boundary of spacetime, does admit conserved charges for metric that are solutions. Our construction is based on background Killing vector…
Minimal Massive Gravity (MMG) is an extension of three-dimensional Topologically Massive Gravity that, when formulated about Anti-de Sitter space, accomplishes to solve the tension between bulk and boundary unitarity that other models in three dimensions suffer from. We study this theory at the chiral point, i.e. at the point of the parameter space where one of the central charges of the dual conformal field theory vanishes. We investigate the non-linear regime of the theory, meaning that we study exact solutions to the MMG field equations that are not Einstein manifolds. We exhibit a large cl…
Sorbonne Université（索邦大学）；Bogazici University（博阿齐奇大学）；Feza Gursey Center for Physics and Mathematics, Bogazici University（博阿齐奇大学费扎·古尔塞物理与数学中心）；Max-Planck-Institut für Gravitationsphysik（马克斯·普朗克引力物理学研究所）；ENSL, CNRS, Laboratoire de physique（里昂高等师范学院、法国国家科学研究中心、物理实验室）；Institut Universitaire de France (IUF)（法兰西学院）
The "Minimal Massive Gravity" (MMG) model of massive gravity in three spacetime dimensions (which has the same anti-de Sitter (AdS) bulk properties as "Topologically Massive Gravity" but improved boundary properties) is coupled to matter. Consistency requires a particular matter source tensor, which is quadratic in the stress tensor. The consequences are explored for an ideal fluid in the context of asymptotically de-Sitter (dS) cosmological solutions, which bounce smoothly from contraction to expansion. Various vacuum solutions are also found, including warped-(A)dS, and (for special values o…
Quantum corrections to the entanglement entropy of matter fields interacting with dynamical gravity have proven to be very important in the study of the black hole information problem. We consider a one-particle excited state of a massive scalar field infalling in a pure AdS$_3$ geometry and compute these corrections for bulk subregions anchored on the AdS boundary. In the dual CFT$_2$, the state is given by the insertion of a local primary operator and its evolution thereafter. We calculate the area and bulk entanglement entropy corrections at order $\mathcal{O}(N^0)$, both in AdS and its CFT…
Under semiclassical evolution, black holes retain a smooth horizon but fail to return information. Yet, the Ryu-Takayanagi prescription computes the boundary entropy expected from unitary CFT evolution. We demonstrate this in a novel setting with an asymptotic bulk detector, eliminating an assumption about the entanglement wedge of auxiliary systems. We consider three interpretations of this result. (i) At face value, information is lost in the bulk but not in the CFT. This conflicts with the AdS/CFT dictionary. (ii) No unique QFT state (pure or mixed) governs all detector responses to the bul…
Designs without disorder: unitary $k$-designs from a single-site bulk defect
University of Geneva（日内瓦大学）；Vrije Universiteit Brussel (VUB)（布鲁塞尔自由大学）；RIKEN Centre for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS)（理化学研究所跨学科理论与数学科学中心）
We obtain a new 3D gravity model from two copies of parity-odd Einstein-Cartan theories. Using Hamiltonian analysis, we demonstrate that the only local degrees of freedom are two massive spin-2 modes. Unitarity of the model in anti-de Sitter and Minkowski backgrounds can be satisfied for vast choices of the parameters without fine-tuning. The recent "exotic massive 3D gravity" model arises as a limiting case of the new model. We also show that there exist trajectories on the parameter space of the new model which cross the boundary between unitary and non-unitary regions. At the crossing point…
The notion of non-Hermitian PT symmetric quantum theory has recently been generalized to the gauge/gravity duality. We study the evolution of such non-Hermitian holographic field theories when the couplings are varied with time with particular emphasis on the question non-unitary time vs. unitary time evolution. We show that a non-unitary time evolution in the dual quantum theory corresponds to a violation of the Null Energy Condition (NEC) in the bulk of the asymptotically AdS spacetime. We find that upon varying the non-Hermitian coupling the horizon of a bulk AdS black hole shrinks. On the …
The exact renormalization group (ERG) for $O(N)$ vector models (at large $N$) on flat Euclidean space can be interpreted as the bulk dynamics corresponding to a holographically dual higher spin gauge theory on $AdS_{d+1}$. This was established in the sense that at large $N$ the generating functional of correlation functions of single trace operators is reproduced by the on-shell action of the bulk higher spin theory, which is most simply presented in a first-order (phase space) formalism. In this paper, we extend the ERG formalism to the wave functionals of arbitrary states of the $O(N)$ vecto…
As an extension of arXiv:\{2002.02448, 2008.10740\} we present a physical protocol that a set of observers can use to detect a pure state in the bulk when they are spread across a small cut near $\mathcal{I}^+_-$ in flat spacetime. The protocol involves the modification of a bulk state using simple unitary operators and measurements of the energy of the state. The states that we study are constructed by acting with low energy operators on a vacuum state such that a perturbative analysis is valid. We restrict ourselves to $3+1$ dimensional spacetimes and only consider massless excitations. From…
If holography is an equivalence between quantum theories, one might expect it to be described by a map that is a bijective isometry between bulk and boundary Hilbert spaces, preserving the hamiltonian and symmetries. Holography has been believed to be a property of gravitational (or string) theories, but not of non-gravitational theories; specifically Marolf has argued that it originates from the gauge symmetries and constraints of gravity. These observations suggest study of the assumed holographic map as a function of the gravitational coupling $G$. The zero coupling limit gives ordinary qua…
We analyse a simple example of a holographically dual pair in which we topologically twist both theories. The holography is based on the two-dimensional N=2 supersymmetric Liouville conformal field theory that defines a unitary bulk quantum supergravity theory in three-dimensional anti-de Sitter space. The supersymmetric version of three-dimensional Liouville quantum gravity allows for a topological twist on the boundary and in the bulk. We define the topological bulk supergravity theory in terms of twisted boundary conditions. We corroborate the duality by calculating the chiral configuration…
For the first time, a gravitational calculation was recently shown to yield the Page curve for the entropy of Hawking radiation, consistent with unitary evolution. However, the calculation takes as essential input Hawking's result that the radiation entropy becomes large at late times. We call this apparent contradiction the state paradox. We exhibit its manifestations in standard and doubly-holographic settings, with and without an external bath. We clarify which version(s) of the Ryu-Takayanagi prescription apply in each setting. We show that the two possible homology rules in the presence o…
Liouville field theory approach to 2-dimensional gravity possesses the duality ($b \leftrightarrow b^{-1}$). The matrix counterpart of minimal gravity $\mathcal{M}(q,p)$ ($q<p$ co-prime) is effectively described on $A_{q-1}$ Frobenius manifold, which may exhibit a similar duality $p\leftrightarrow q$, and allow a description on $A_{p-1}$ Frobenius manifold. We have positive results from the bulk one-point and the bulk-boundary two-point correlations on disk that the dual description of the Frobenius manifold works for the unitary series $\mathcal{M}(q, q+1)$. However, for the Lee-Yang series $…
We consider the problem of defining spacelike-supported boundary-to-bulk propagators in AdS$_{d+1}$ down to the unitary bound $Δ=(d-2)/2$. That is to say, we construct the `smearing functions' $K$ of HKLL but with different boundary conditions where both dimensions $Δ_+$ and $Δ_-$ are taken into account. More precisely, we impose Robin boundary conditions, which interpolate between Dirichlet and Neumann boundary conditions and we give explicit expressions for the distributional kernel $K$ with spacelike support. This flow between boundary conditions is known to be captured in the boundary by a…
We construct and study an ensemble of non-isometric error correcting codes in a toy model of an evaporating black hole in two-dimensional dilaton gravity. In the preferred bases of Euclidean path integral states in the bulk and Hamiltonian eigenstates in the boundary, the encoding map is proportional to a linear transformation with independent complex Gaussian random entries of zero mean and unit variance. Using measure concentration, we show that the typical such code is very likely to preserve pairwise inner products in a set $S$ of states that can be subexponentially large in the microcanon…
We consider the space-time at short distances in which it is described by a $D$-dimensional manifold (bulk) carrying out the principal bundle structure. As a result, this space-time manifold is foliated in the covariant way by the $(D-4)$-dimensional submanifolds, realized as the space-like internal spaces, that are smooth copies of the Lie group $G$ considered in this paper as the special unitary group. The submanifolds being transversal to the internal spaces are realized as the external spaces and in fact identified as the usual $4$-dimensional world. The fundamental degrees of freedom dete…
Unitary and Nonunitary A-D-E minimal models: Coset graph fusion algebras, defects, entropies, SREEs and dilogarithm identities
School of Mathematics and Statistics, University of Melbourne（墨尔本大学数学与统计学院）；School of Mathematics and Physics, University of Queensland（昆士兰大学数学与物理学院）；High Energy Physics Research Unit, Faculty of Science Chulalongkorn University（朱拉隆功大学理学院高能物理研究单元）
