We calculate the four-wave scattering amplitude in the background of an AdS traversable wormhole in 2+1 dimensions created by a nonlocal coupling of AdS boundaries in the BTZ black hole background. The holographic dual of this setup is a pair of CFTs coupled via a double-trace deformation, the scattering amplitude giving the out-of-time ordered correlator (OTOC) in CFT. Short-living wormholes always exhibit a regime of fast scrambling, saturating the MSS bound for the Lyapunov exponent but in the early-time regime the scrambling can be slower, with a Lyapunov exponent linear in $T$ but below the MSS bound. For long-living (near-eternal) wormholes however the numerics suggests the existence of another regime, with a drastic (exponential) slowdown of scrambling and an exponentially small Lyapunov exponent. Our findings have parallels in the SYK model, and may indicate certain limitations of wormhole teleportation protocols previously studied in the literature.
We calculate the four-wave scattering amplitude in the background of an AdS traversable wormhole in 2+1 dimensions created by a nonlocal coupling of AdS boundaries in the BTZ black hole background. The holographic dual of this setup is a pair of CFTs coupled via a double-trace deformation, the scattering amplitude giving the out-of-time ordered correlator (OTOC) in CFT. Short-living wormholes always exhibit a regime of fast scrambling, saturating the MSS bound for the Lyapunov exponent but in the early-time regime the scrambling can be slower, with a Lyapunov exponent linear in T but below the MSS bound. For long-living (near-eternal) wormholes however the numerics suggests the existence of another regime, with a drastic (exponential) slowdown of scrambling and an exponentially small Lyapunov exponent. Our findings have parallels in the SYK model, and may indicate certain limitations of wormhole teleportation protocols previously studied in the literature.
Abstract We compute out-of-time-order correlators (OTOCs) in two-dimensional holographic conformal field theories (CFTs) with different left- and right-moving temperatures. Depending on whether the CFT lives on a spatial line or circle, the dual bulk geometry is a boosted BTZ black brane or a rotating BTZ black hole. In the case when the spatial direction is non-compact, we generalise a computation of Roberts and Stanford and show that to reproduce the correct bulk answer a maximal channel contribution needs to be selected when using the identity block approximation. We use the correspondence between global conformal blocks and geodesic Witten diagrams to extend our results to CFTs on a spatial circle. In [1] it was shown that the OTOC for a rotating BTZ black hole exhibits a periodic modulation about an average exponential decay with Lyapunov exponent 2π/β. In the extremal limit where the black hole is maximally rotating, it was shown in [2] that the OTOC exhibits an average cubic growth, on which is superposed a sawtooth pattern which has small periods of Lyapunov growth due to the non-zero temperature of left-movers in the dual CFT. Our computations explain these results from a dual CFT perspective.
Fast scrambling and exponential growth of complexity are two distinctive features of quantum gravity, offering a unique window into the mechanisms that govern the semiclassical limit of the holographic duality. In this thesis, we explore these information-theoretic phenomena through two key diagnostics --- out-of-time ordered correlation functions (OTOCs) and spread complexity --- computed in the setting of two-dimensional holographic conformal field theories (CFTs). First, we focus on the large-c limit of Liouville theory, where the computation of OTOCs reveals distinctive insights into both the chaotic nature and the structural features of the theory. Remarkably, we find it exhibits a maximal Lyapunov exponent, despite lacking the identity operator in its spectrum, and a scrambling time logarithmic in the central charge. Using path integral techniques, we calculate heavy-light four-point functions, and uncover a conformal block expansion dominated by the exchange of an infinite tower of spinless double-trace operators. We attribute the chaotic behaviour of the OTOC to this exchange of operators, which can be thought of as dressed scramblons. Additionally, we observe that at the onset of scrambling, the dominance of conformal blocks shifts from heavy primaries at early times to light primaries at late times. Second, we turn to the diagnostic of spread complexity in locally excited holographic CFTs and argue that its time derivative is dual to the proper radial momentum of a massive point particle propagating in Anti-de Sitter spacetime (AdS). Exploiting the conformal symmetry and the emergence of an ``effective'' SL(2,R) group, we employ representation-theoretic techniques to compute return amplitudes, Lanczos coefficients, and spread complexities. We then verify our conjectured duality for three settings: the CFT on the line, dual to Poincaré AdS; the CFT on the circle, dual to global AdS; and the CFT at finite temperature, dual to a BTZ black hole. In the gravitational theory, the same SL(2,R) structure emerges as the isometry group of the infalling plane of the bulk particle. Third, we generalise the analysis of spread complexity to modular Hamiltonian evolution. Exploiting the rich algebraic structure of modular theory, we construct the modular Hamiltonians Kₘ and the half-sided modular translations G for the three example states above. Leveraging conformal symmetry, we once again compute modular return amplitudes, Lanczos coefficients, and modular spread complexities. In particular, we emphasise the role of ``conformal regularisation'' --- i.e. regulating with the operator that generates modular evolution - in preserving the effective SL(2,R) structure of the amplitudes. We find that evolution under Kₘ exhibits thermal spread complexity for β = 1 across all examples, while G-evolution universally leads to quadratic growth. Finally, we provide a formal explanation of these results, identifying the role of the Killing norm D of the evolution operator as the key factor governing the behaviour of complexity.
In the Sachdev-Ye-Kitaev (SYK) model, traversable wormhole teleportation fidelity probes the many-body scrambling of holographic black holes. We apply a gravitational-wave-inspired periodic Floquet deformation derived from the lowest-dimensional bilinear SYK4 sector to the boundary, thereby characterizing the channel response via exact numerical time evolution at βJ=2. Re-optimizing under the drive isolates genuine physical effects from calibration mismatch, yielding four main results: (i) distinct perturbative and non-perturbative amplitude regimes separated near ε∼J; (ii) a natural low-pass filter response, with maximum fidelity suppression at ω≲β−1 and monotonic recovery above the thermal scale; (iii) a scrambling delay, evidenced by a delayed fidelity peak under an inspiral chirp (Δtscr(fid)=+0.11J−1) and independently confirmed by OTOC measurements to grow monotonically with drive amplitude; and (iv) robust, non-zero fidelity suppression across N∈{10,12,14,16} Majorana modes. These findings establish that holographic wormholes degrade gracefully under metric-like boundary deformations, providing direct diagnostic signatures for near-term quantum hardware.
Abstract Recent gravity discussions of a traversable wormhole indicate that in holographic systems signals generated by a source could reappear long after they have dissipated, with the need of only performing some simple operations. In this paper we argue the phenomenon, to which we refer as “regenesis”, is universal in general quantum chaotic many-body systems, and elucidate its underlying physics. The essential elements behind the phenomenon are: (i) scrambling which in a chaotic system makes out-of-time-ordered correlation functions (OTOCs) vanish at large times; (ii) the entanglement structure of the state of the system. The latter aspect also implies that the regenesis phenomenon requires fine tuning of the initial state. Compared to other manifestations of quantum chaos such as the initial growth of OTOCs which deals with early times, and a random matrix-type energy spectrum which reflects very large time behavior, regenesis concerns with intermediate times, of order the scrambling time of a system. We also study the phenomenon in detail in general two-dimensional conformal field theories in the large central charge limit, and highlight some interesting features including a resonant enhancement of regenesis signals near the scrambling time and their oscillations in coupling. Finally, we discuss gravity implications of the phenomenon for systems with a gravity dual, arguing that there exist regimes for which traversability of a wormhole is quantum in nature, i.e. cannot be associated with a semi-classical spacetime causal structure.
Four coupled SYK models and Nearly AdS2 gravities: Phase Transitions in Traversable wormholes and in Bra-ket wormholes
We study four coupled SYK models and nearly AdS2 gravities. In the SYK model side, we construct a model that couples two copies of two coupled SYK models. In nearly AdS2 gravity side, we entangle matter fields in two copies of traversable wormholes. In both cases, the systems show first order phase transitions at zero temperature by changing couplings, which is understood as the exchange of traversable wormhole configurations. In nearly AdS2 gravity cases, by exchanging the role of space and time the wormholes are interpreted as bra-ket wormholes. In Lorentzian signature, these bra-ket wormholes lead to two closed universes that are entangled with each other as well as matter fields in the flat space without dynamical gravity. We study the effect of projection or entangling operation for matters on flat spaces and they cause phase transitions in bra-ket wormholes, which leads to the pair annihilation of closed universes. Using these bra-ket wormholes, we discuss the way to embed states in 2d holographic CFTs into Hilbert space of many 2d free fields.
A bstract In this note, we investigate the out-of-time-order correlators (OTOCs) for quantum fields in a holographic framework describing Einstein-Podolsky-Rosen (EPR) pairs. We compute the four-point and six-point OTOCs using the gravity dual, represented by the string worldsheet theory in Anti-de Sitter (AdS) space. These correlators quantify the rate at which information is scrambled, leading to the disentanglement of the EPR pair. We demonstrate consistency between two approaches for calculating OTOCs: the holographic influence functional on worldsheets perturbed by shock waves, and the worldsheet scattering in the eikonal approximation. We show that the OTOCs exhibit an initial phase of exponential growth, with six-point correlators indicating a marginally longer scrambling time compared to four-point correlators.