In this paper, we obtain a bulk dual to SYK model, including SYK model with U (1) charge, by Kaluza-Klein (KK) reduction from three dimensions. We show that KK reduction of the 3D Einstein action plus its boundary term gives the Jackiw-Teitelboim (JT) model in 2D with the appropriate 1D boundary term. The size of the KK radius gets identified with the value of the dilaton in the resulting near-AdS2 geometry. In presence of U(1) charge, the 3D model additionally includes a U (1) Chern-Simons (CS) action. In order to describe a boundary theory with non-zero chemical potential, we also introduce a coupling between CS gauge field and bulk gravity. The 3D CS action plus the new coupling term with appropriate boundary terms reduce in two dimensions to a BF-type action plus a source term and boundary terms. The KK reduced 2D theory represents the soft sector of the charged SYK model. The pseudo-Nambu-Goldstone modes of combined Diff/SL(2, R) and U (1) local /U (1) transformations are represented by combined large diffeomorphisms and large gauge transformations. The effective action of the former is reproduced by the action cost of the latter in the bulk dual, after appropriate identification of parameters. We compute chaotic correlators from the bulk and reproduce the result that the contribution from the "boundary photons" corresponds to zero Liapunov exponent.

We study the entanglement for a state on linked torus boundaries in 3d ChernSimons theory with a generic gauge group and present the asymptotic bounds of Rényi entropy at two different limits: (i) large Chern-Simons coupling k, and (ii) large rank r of the gauge group. These results show that the Rényi entropies cannot diverge faster than ln k and ln r, respectively. We focus on torus links T (2, 2n) with topological linking number n. The Rényi entropy for these links shows a periodic structure in n and vanishes whenever n = 0 (mod p), where the integer p is a function of coupling k and rank r. We highlight that the refined Chern-Simons link invariants can remove such a periodic structure in n.

We develop a new method for computing the holographic retarded propagator in generic (non-)equilibrium states using the state/geometry map. We check that our method reproduces the thermal spectral function given by the Son-Starinets prescription. The time-dependence of the spectral function of a relevant scalar operator is studied in a class of non-equilibrium states. The latter are represented by AdS-Vaidya geometries with an arbitrary parameter characterising the timescale for the dual state to transit from an initial thermal equilibrium to another due to a homogeneous quench. For long quench duration, the spectral function indeed follows the thermal form at the instantaneous effective temperature adiabatically, although with a slight initial time delay and a bit premature thermalisation. At shorter quench durations, several new non-adiabatic features appear: (i) time-dependence of the spectral function is seen much before than that in the effective temperature (advanced time-dependence), (ii) a big transfer of spectral weight to frequencies greater than the initial temperature occurs at an intermediate time (kink formation) and (iii) new peaks with decreasing amplitudes but in greater numbers appear even after the effective temperature has stabilised (persistent oscillations). We find four broad routes to thermalisation for lower values of spatial momenta. At higher values of spatial momenta, kink formations and persistent oscillations are suppressed, and thermalisation time decreases. The general thermalisation pattern is globally top-down, but a closer look reveals complexities.

SYK model is a quantum mechanical model of fermions which is solvable at strong coupling and plays an important role as perhaps the simplest holographic model of quantum gravity and black holes. The present work considers a deformed SYK model and a sudden quantum quench in the deformation parameter. The system, as in the undeformed case, permits a low energy description in terms of pseudo Nambu Goldstone modes. The bulk dual of such a system represents a gravitational collapse, which is characterized by a bulk matter stress tensor whose value near the boundary shows a sudden jump at the time of the quench. The resulting gravitational collapse forms a black hole only if the deformation parameter ∆ exceeds a certain critical value ∆ c and forms a horizonless geometry otherwise. In case a black hole does form, the resulting Hawking temperature is given by a fractional power T bh ∝ (∆ − ∆ c) 1/2 , which is reminiscent of the 'Choptuik phenomenon' of critical gravitational collapse.The black hole information loss problem is usually associated with black hole evaporation. However, even the process of formation of a black hole can be regarded as a version of information loss. This is because even if the collapsing matter is in a pure state, when it forms a black hole it has an entropy. Hence a pure state appears to evolve to a mixed (thermal) state; furthermore, the information about the initial pure state appears to be lost. How does one understand this puzzle within a unitary quantum mechanical framework? With the AdS/CFT correspondence, such a unitary description appears possible in terms of the dual CFT where gravitational collapse can be modelled by a quantum quench [1][2][3][4] and under a sudden perturbation a given pure state can evolve to a pure state with thermal properties 1 . Such models are not easy to construct in strongly coupled field theories in three and higher dimensions. In lower dimensions, however, there are powerful techniques to deal with the dynamics of strongly coupled conformal field theories. In one dimension, the relevant strongly coupled model [5][6][7][8][9][10] which has a holographic dual [11][12][13][14] is the SYK model 2 . In the present paper, we will discuss gravitational collapse in such a holographic dual.Besides the above issue of 'information loss', gravitational collapse is associated with another interesting phenomenon, namely that of Choptuik scaling. In his classic work [18] Choptuik analyzed a family of initial states characterized by a parameter p (which roughly corresponds to the amount of self-gravitation of the infalling matter) and evolved them numerically (see, e.g.[19] for a review). He found that while no black holes are formed for p < p c , they are formed for p > p c , with the mass of the resulting black hole given byHere, γ is found to be a universal critical exponent, which depends only on the type of infalling matter and not on the details of the initial configuration. The results of [18] were in asymptotically flat space (for a review see, e.g. [20]). This wa...