We show that the holographic description of a class of AdS black holes with scalar hair involves dual field theories with a double well effective potential. Black hole microstates have significant support around both vacua in the dual, which correspond to perturbative degrees of freedom on opposite sides of the horizon. A solvable toy-model version of this dual is given by a quantum mechanical particle in a double well potential. In this we show explicitly that the interactions replace the state-dependence that is needed to describe black hole microstates in a low energy effective model involving the tensor product of two decoupled harmonic oscillators. A naive number operator signals the presence of a firewall but a careful construction of perturbative states and operators extinguishes this.
本文研究了谐振子势中单个厄米矩阵的大N规范量子力学，并作为AdS/CFT对应的玩具模型.我们认为，对偶几何应该是一个字符串在二维的曲率stringy大小。即使对偶几何不是弱弯曲的，人们仍然可以从开闭弦对偶的详细研究中获得系统的知识。我们给出了由迹（闭弦）构成的状态基与矩阵（D-膜图像）的本征值之间的Schur多项式映射。我们把这个模型与AdS_5 x S^5中巨引力子的研究联系起来。我们证明了在矩阵模型中，两个沿AdS_5和S^5扩展的巨引力子可以分别解释为从费米海中取一个本征值并对其进行极大的激发，或者在费米海中开一个洞.这与最近对c=1弦的研究相似。这种联系为如何计算巨型引力子提供了新的见解。
The holographic duals of Entanglement of Purification through the Entanglement Wedge Cross Section has been a well-discussed topic in the literature recently. More general entanglement measures involving multipartite information and their holographic duals have also been proposed. On the other hand, the recent intriguing program reproducing the Page Curve in Black hole entropy using the notion of islands has also been an obvious issue of attraction. A toy model involving Multiboundary wormholes in AdS$_{3}$ was able to capture many interesting facts about such calculations. In such a toy model…
We extend our previous study of a toy model for coupling classical Yang-Mills equations for describing overoccupied gluons at the saturation scale with a strongly coupled infrared sector modeled by AdS/CFT. Including propagating modes in the bulk we find that the Yang-Mills sector loses its initial energy to a growing black hole in the gravity dual such that there is a conserved energy-momentum tensor for the total system while entropy grows monotonically. This involves a numerical AdS simulation with a backreacted boundary source far from equilibrium.
Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence
我们提出了一个家庭的AdS/CFT对应的一种新的构造的量子纠错码的张量网络结构的基础上，完全可解的玩具模型。我们的构建块是一种特殊类型的张量与最大的纠缠沿任何bipartition，这引起了一个等距从散装希尔伯特空间的边界希尔伯特空间。整个张量网络是量子纠错码的编码器，其中体自由度和边界自由度可以分别被标识为逻辑自由度和物理自由度。这些模型捕捉到了AdS/CFT对应中纠缠的关键特征;特别是，Ryu-Takayanagi公式和三方信息的负性在许多情况下都完全遵守。体逻辑算子可以在多个边界区域上表示，这模拟了从体算子重构边界算子的Rindler-wedge重构，明确实现了Almheiri等人最近提出的AdS/CFT的量子纠错特性. al in arXiv：1411.7041.
We present a general procedure for constructing tensor networks that accurately reproduce holographic states in conformal field theories (CFTs). Given a state in a large-$N$ CFT with a static, semiclassical gravitational dual, we build a tensor network by an iterative series of approximations that eliminate redundant degrees of freedom and minimize the bond dimensions of the resulting network. We argue that the bond dimensions of the tensor network will match the areas of the corresponding bulk surfaces. For 'tree' tensor networks (i.e., those that are constructed by discretizing spacetime wit…
In this work we reason that there is a universal picture for several different holographic toy model constructions, and a gravity-like bulk field theory that gives rise it. First, we observe that the perfect tensor-networks and hyperbolic fracton models are both equivalent to the even distribution of bit-threads on geodesics in the AdS space. Such picture is also a natural "leading-order" approximation to the holographic entanglement properties. Then, we argue that the rank-2 U(1) theory with linearized diffeomorphism as its gauge symmetry, also known as a case of Lifshitz gravity, is the bulk…
The paper is another step towards a realisation of the goal, advanced in articles 1706.05682 [hep-th] and 1808.04470 [hep-th], of a systematic supersymmetry-equivariant geometrisation of physically distinguished Green-Schwarz super-(p+2)-cocycles defining classes in the supersymmetry-invariant refinement of the de Rham cohomology of homogeneous spaces of (supersymmetry) Lie supergroups, associated with reductive decompositions of their Lie superalgebras. It deals with a correlated geometrisation of a pair of such super-(p+2)-cocycles on spaces in correspondence under a blow-up transformation d…
We study a toy model of the Kerr/CFT correspondence using string theory on AdS$_3 \times S^3$. We propose a single trace irrelevant deformation of the dual CFT generated by a vertex operator with spacetime dimensions (2,1). This operator shares the same quantum numbers as the integrable $T\bar{J}$ deformation of two-dimensional CFTs where $\bar{J}$ is a chiral $U(1)$ current. We show that the deformation is marginal on the worldsheet and that the target spacetime is deformed to null warped AdS$_3$ upon dimensional reduction. We also calculate the spectrum of the deformed theory on the cylinder…
We consider the algebra of simple operators defined in a time band in a CFT with a holographic dual. When the band is smaller than the light crossing time of AdS, an entire causal diamond in the center of AdS is separated from the band by a horizon. We show that this algebra obeys a version of the Reeh-Schlieder theorem: the action of the algebra on the CFT vacuum can approximate any low energy state in the CFT arbitrarily well, but no operator within the algebra can exactly annihilate the vacuum. We show how to relate local excitations in the complement of the central diamond to simple operat…
These lecture notes are based on a series of lectures given at the XIII Modave summer school in mathematical physics. We review the construction due to Hamilton, Kabat, Lifschytz and Lowe for reconstructing local bulk operators from CFT operators in the context of AdS/CFT and show how to recover bulk correlation functions from this definition. Building on the work of these authors, it has been noted that the bulk displays quantum error correcting properties. We will discuss tensor network toy models to exemplify these remarkable features. We will discuss the role of gauge invariance and of dif…
${\cal N} = 8$ superconformal field theories, such as the ABJM theory at Chern-Simons level $k=1$ or $2$, contain 35 scalar operators ${\cal O}_{IJ}$ with $Δ=1$ in the ${\bf 35}_v$ representation of SO(8). The 3-point correlation function of these operators is non-vanishing, and indeed can be calculated non-perturbatively in the field theory. But its AdS$_4$ gravity dual, obtained from gauged ${\cal N}=8$ supergravity, has no cubic $A^3$ couplings in its Lagrangian, where $A^{IJ}$ is the bulk dual of ${\cal O}_{IJ}$. So conventional Witten diagrams cannot furnish the field theory result. We sh…
A Geodesic Route toward Holography beyond AdS: Tessellating the Schwarzschild Black Hole
School of Physics and Shing-Tung Yau Center, Southeast University（东南大学物理学院与邵逸夫数学中心）；College of Computer Science and Technology, Zhejiang University（浙江大学计算机科学与技术学院）
The AdS/CFT correspondence realises the holographic principle where information in the bulk of a space is encoded at its border. We are yet a long way from a full mathematical construction of AdS/CFT, but toy models in the form of holographic quantum error correcting codes (HQECC) have replicated some interesting features of the correspondence. In this work we construct new HQECCs built from random stabilizer tensors that describe a duality between models encompassing local Hamiltonians whilst exactly obeying the Ryu-Takayanagi entropy formula for all boundary regions. We also obtain complemen…
We discuss information loss from black hole physics in AdS$_3$, focusing on two sharp signatures infecting CFT$_2$ correlators at large central charge $c$: 'forbidden singularities' arising from Euclidean-time periodicity due to the effective Hawking temperature, and late-time exponential decay in the Lorentzian region. We study an infinite class of examples where forbidden singularities can be resolved by non-perturbative effects at finite $c$, and we show that the resolution has certain universal features that also apply in the general case. Analytically continuing to the Lorentzian regime, …
We investigate feasible holography with the Kitaev model using dilatonic gravity in AdS$_2$. We propose a generic dual theory of gravity in the AdS$_2$ and suggest that this bulk action is a suitable toy model in studying quantum mechanics in the Kitaev model using gauge/gravity duality. This gives a possible equivalent description for the Kitaev model in the dual gravity bulk. Scalar and tensor perturbations are investigated in detail. In the case of near AdS perturbation, we show that the geometry still "freezes" as is AdS, while the dilation perturbation decays at the AdS boundary safely. T…
Motivated by the recent progress in solving the large charge sector of conformal field theories, we revisit the mass-charge relation of boson stars asymptotic to global AdS. We construct and classify a large number of electrically charged boson star solutions in a toy model and two supergravity models arising from the $SU(3)$ and $U(1)^4$ truncation of $D=4$ $SO(8)$ gauged maximal supergravity. We find a simple ansatz for the chemical potential that can fit the numerical data in striking accuracy for the full range of charge. Combining with the first law of thermodynamics, we can then evaluate…
In 2013, Balasubramanian presented a 5+1 dimensional holographic toy model that allows for an exact solution to Einstein's equations in the bulk in which the isometries of $AdS_5$ appear to be broken to an isometry group describing a discretely scale invariant and Poincaré invariant setup [arXiv:1301.6653]. In this paper, we investigate this solution in more detail. By analytically solving the Killing equations, we prove that the full $AdS_5$ isometry group is still present, although in a somewhat hidden way. We will also comment on the prospects of finding other holographic bottom up toy mode…
Motivated by the quantum process of black hole evaporation and its implications for symmetries, we consider a qubit system with a random dynamics as a toy model of black hole. We compute its symmetry-resolved entropies and discuss its implications. We first consider the case where charges are conserved and compute the symmetry-resolved entropies. We derive a symmetry-resolved analogue of the Page curve. We then consider the case where symmetry is explicitly broken and charges are no longer conserved. It serves as a toy model for global symmetry breaking in black hole evaporation. Despite the s…
In this work, we investigate how single-sided and eternal black holes in AdS can host an enormous number of semiclassical excitations in their interior, which is seemingly not reflected in the Bekenstein Hawking entropy. In addition to the paradox in the entropy, we argue that the treatment of such excitations using effective field theory also violates black holes' expected spectral properties. We propose that these mysteries are resolved because apparently orthogonal semiclassical bulk excitations have small inner products between them; and consequently, a vast number of semiclassical excitat…
We explore the Hawking evaporation of two-dimensional anti-de Sitter (AdS$_2$), dilatonic black hole coupled with conformal matter, and derive the Page curve for the entanglement entropy of radiation. We first work in a semiclassical approximation with backreaction. We show that the end-point of the evaporation process is AdS$_2$ with a vanishing dilaton, i.e. a regular, singularity-free, zero-entropy state. We explicitly compute the entanglement entropies of the black hole and the radiation, as functions of the horizon radius, using the conformal field theory (CFT) dual to AdS$_2$ gravity. We…
We revisit the computation of the 1-loop string correction to the "latitude" minimal surface in $AdS_5 \times S^5$ representing 1/4 BPS Wilson loop in planar $\cal N$=4 SYM theory previously addressed in arXiv:1512.00841 and arXiv:1601.04708. We resolve the problem of matching with the subleading term in the strong coupling expansion of the exact gauge theory result (derived previously from localization) using a different method to compute determinants of 2d string fluctuation operators. We apply perturbation theory in a small parameter (angle of the latitude) corresponding to an expansion nea…
We review the paradox of low energy excitations about an AdS black hole. An appropriately chosen unitary operator in the boundary theory can create a locally strong excitation near the black hole horizon, whose global energy is small as a result of the gravitational redshift. The paradox is that this seems to violate a general rule of statistical mechanics, which states that an operator with energy parametrically smaller than $k T$ cannot create a significant excitation in a thermal system. When we carefully examine the position dependence of the boundary unitary operator that produces the exc…
We employ the "complexity equals action" conjecture to investigate the action growth rate for the charged and neutral AdS black branes of a holographic toy model consisting of Einstein-Maxwell theory in $d + 1$-dimensional bulk spacetime with $d - 1$ massless scalar fields which is called Einstein-Maxwell-Axion (EMA) theory. From the holographic point of view, the scalar fields source a spatially dependent field theory with momentum relaxation on the boundary, which is dual to the homogeneous and isotropic black branes. We find that the growth rate of the holographic complexity within the Whee…
We test the proposal of arXiv:2012.11274 for the holographic computation of the charged moments and the resulting symmetry-resolved entanglement entropy in different excited states, as well as for two entangling intervals. Our holographic computations are performed in $U(1)$ Chern-Simons-Einstein-Hilbert gravity, and are confirmed by independent results in a conformal field theory at large central charge. In particular, we consider two classes of excited states, corresponding to charged and uncharged conical defects in AdS${}_3$. In the conformal field theory, these states are generated by the…
To find the origin of chaos near black hole horizon in string-theoretic AdS/CFT correspondence, we perform a chaos analysis of a suspended string in AdS black hole backgrounds. It has a definite CFT interpretation: chaos of Wilson loops, or in other words, sensitive time-evolution of a quark antiquark force in thermal gauge theories. Our nonlinear numerical simulation of the suspended Nambu-Goto string shows chaos, which would be absent in pure AdS background. The calculated Lyapunov exponent $λ$ satisfies the universal bound $λ\leq 2πT_{\rm H}$ for the Hawking temperature $T_{\rm H}$. We also…
We numerically investigate spherically symmetric collapses in the Gross-Pitaevskii equation with attractive nonlinearity in a harmonic potential. Even below threshold for direct collapse, the wave function bounces off from the origin and may eventually become singular after a number of oscilla- tions in the trapping potential. This is reminiscent of the evolution of Einstein gravity sourced by a scalar field in Anti-de Sitter space where collapse corresponds to black hole formation. We carefully examine the long time evolution of the wave function for continuous families of initial states in o…
We study Hartle-Hawking-like states of quantum field theories on asymptotically AdS black hole backgrounds, with particular regard to the phase structure of interacting theories. By a suitable analytic continuation we show that the equilibrium dynamics of field theories on large asymptotically AdS black holes can be related to the low temperature states of the same field theory on the AdS soliton (or pure AdS) background. This allows us to gain insight into Hartle-Hawking-like states on large-radius Schwarzschild- or rotating-AdS black holes. Furthermore, we exploit the AdS/CFT correspondence …
Einsteinian cubic gravity provides a holographic toy model of a nonsupersymmetric CFT in three dimensions, analogous to the one defined by Quasi-topological gravity in four. The theory admits explicit non-hairy AdS$_4$ black holes and allows for numerous exact calculations, fully nonperturbative in the new coupling. We identify several entries of the AdS/CFT dictionary for this theory, and study its thermodynamic phase space, finding interesting new phenomena. We also analyze the dependence of Rényi entropies for disk regions on universal quantities characterizing the CFT. In addition, we show…
Lovelock theory of gravity provides a tractable model to investigate the effects of higher-curvature terms in the context of AdS/CFT. Yielding second order, ghost-free field equations, this theory represents a minimal setup in which higher-order gravitational couplings in asymptotically Anti-de Sitter (AdS) spaces, including black holes, can be solved analytically. This however has an obvious limitation as in dimensions lower than seven, the contribution from cubic or higher curvature terms is merely topological. Therefore, in order to go beyond quadratic order and study higher terms in AdS_5 …
