AdS$_2$ holography and the SYK model

Gábor Sárosi

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AdS$_2$ holography and the SYK model

Gábor Sárosi

arXiv:1711.08482v6Nov 22, 2017hep-thcond-mat.str-elgr-qc

TL;DR

AdS2 holography is difficult because excitations backreact on the asymptotic geometry and near-extremal black holes possess a gap obstructing complete decoupling. These notes review cutoff AdS2 and Jackiw-Teitelboim theory, then introduce SYK and its relation to near-horizon black-hole physics. They also examine Schwarzian dynamics, large-q thermodynamics, four-point functions, chaos, and proposed bulk interpretations, while identifying limitations in the fluctuation and bulk-dual analyses.

Problem
AdS2 holography must account for the destruction of asymptotic AdS2 by excitations and the unusual noncommuting limits relevant to near-extremal black holes.
Method
The notes review AdS2 geometry and dilaton-gravity backreaction, derive cutoff Jackiw-Teitelboim dynamics, and introduce SYK through large-q, Schwarzian, four-point, and chaos analyses.
Results
The notes identify universal low-energy dynamics in cutoff AdS2 and discuss SYK reparametrization modes, Schwarzian behavior, thermal entropy, four-point functions, and chaos.
Takeaways & Limitations
The notes connect SYK's low-energy structure to features of near-horizon regions of near-extremal black holes, motivating its study as a microscopic model.
Takeaways & Limitations
The proposed holographic bulk interpretation is too naive because the resulting Laplacian is Lorentzian despite starting from a Euclidean boundary theory.

Abstract

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These are lecture notes based on a series of lectures presented at the XIII Modave Summer School in Mathematical physics aimed at PhD students and young postdocs. The goal is to give an introduction to some of the recent developments in understanding holography in two bulk dimensions, and its connection to microscopics of near extremal black holes. The first part reviews the motivation to study, and the problems (and their interpretations) with holography for AdS$_2$ spaces. The second part is about the Jackiw-Teitelboim theory and nearly-AdS$_2$ spaces. The third part introduces the Sachdev-Ye-Kitaev model, reviews some of the basic calculations and discusses what features make the model exciting.

1 Foreword

These lecture notes introduce AdS2 holography, near-extremal black-hole physics, Jackiw-Teitelboim theory, and the SYK model for readers entering the subject.

The notes review holography in AdS2 and its connection to black-hole physics, emphasizing universal deep-infrared dynamics described by Jackiw-Teitelboim theory.They then overview gravitational dynamics and matter coupling in this model.
The SYK model is a quantum-mechanical system of N Majorana fermions with all-to-all random couplings.The notes introduce the model and discuss features shared with near-horizon regions of near-extremal black holes.
The SYK section is mostly readable independently, while the first two parts assume familiarity with higher-dimensional AdS/CFT.The notes point readers to an external review for that background.
The notes are intended to provide basic familiarity with these developments to readers who did not attend the lectures.The author warns that the notes contain errors and welcomes comments and corrections.

2 Motivation

The motivation for AdS2 holography comes from near-horizon limits of extremal black holes, but backreaction and the black-hole gap obstruct a straightforward decoupling limit. Cutoff AdS2 and Jackiw-Teitelboim theory isolate universal infrared dynamics.

2.1 Near horizon region of extremal black holes: Extremal Reissner-Nordström near-horizon geometry becomes AdS2 × S2, with the AdS2 factor expressed in Poincaré coordinates.The construction zooms into the horizon while taking ℓP →0 with a rescaled coordinate fixed.
2.1 Near horizon region of extremal black holes: Global AdS2 has two distinct boundaries, while the Poincaré and Rindler patches cover different regions of its causal diagram.The Penrose diagram displays these coordinate regions and the near-horizon AdS2 strip.
2.2 The backreaction problem: Any excitation backreacts strongly enough to destroy the assumed asymptotic AdS2 geometry in the relevant dilaton-gravity models.A nonzero matter stress tensor requires the dilaton to diverge near at least one boundary.
2.2.1 Black hole gap: The near-extremal black-hole gap diverges as ℓP →0, so the strict near-horizon limit retains only extremal ground states rather than excitations.Keeping finite excitation energy, charge, and Hawking temperature simultaneously is impossible in this limit.
2.3 Holographic interpretation: To access excitations, one must zoom out from complete decoupling, leaving coupling to the asymptotic region through an irrelevant RG deformation.The deep infrared nevertheless remains close to the AdS2 fixed point.
2.4 Jackiw-Teitelboim theory: With suitable cutoff and dilaton boundary conditions, the interior dynamics is largely universal and is governed by Jackiw-Teitelboim theory.The broader dilaton-gravity family supplies the ultraviolet completion of the cutoff AdS2 space.

3 Nearly AdS2 spaces

Nearly-AdS2 configurations reduce to hyperbolic-disk cutouts whose reparametrization zero modes are explicitly lifted by the dilaton, yielding Schwarzian boundary dynamics and thermal entropy linear in temperature.

3.1 Euclidean Jackiw-Teitelboim: The nearly-AdS2 configuration space consists of differently shaped cutouts of the hyperbolic disk, with boundary trajectory t(u) as its dynamical variable.The induced-metric condition fixes the parametrization up to a single function t(u).
3.1 Euclidean Jackiw-Teitelboim: Euclidean AdS2 is the hyperbolic disk; Poincaré time is noncompact, whereas Euclidean Rindler time is a 2π-periodic angular coordinate.Both Euclidean coordinate systems cover the entire disk.
3.1 Euclidean Jackiw-Teitelboim: The Einstein-Hilbert term is topological and gives every simply connected hyperbolic-disk cutout the same action, leaving shape deformations as zero modes.The dilaton-dependent term is introduced to lift this degeneracy.
3.2 Configuration space: Boundary reparametrizations are spontaneously broken to an SL(2,R) subgroup, so t(u) acts as a one-dimensional Goldstone mode with zero action.Translations and rotations preserve a fixed cutout and form the unbroken subgroup.
3.3 Schwarzian theory: The dilaton explicitly breaks reparametrization symmetry and reduces Jackiw-Teitelboim gravity to a Schwarzian boundary theory.This breaking represents departure from the very-near-horizon region toward the UV completion.
3.3.1 Thermal entropy: Thermal solutions use periodic Euclidean Rindler time, so the boundary period β is interpreted as the inverse temperature; the resulting entropy is linear in temperature.The thermal entropy is interpreted as the entropy of a near-extremal black hole.

3.4 Coupling to matter

Coupling a bulk scalar to nearly-AdS2 gravity makes matter correlators depend on the boundary reparametrization, while operator ordering controls the structure of gravitationally corrected four-point functions.

3.4 Coupling to matter: A free massive bulk scalar couples to the Schwarzian mode, and its boundary source is defined through the near-boundary coefficient χ(z,t) = z^(1−∆)χ̃r(t) + ···.The source is interpreted as coupling to an operator of scaling dimension ∆.
3.4 Coupling to matter: In two dimensions, the scalar partition function depends on the boundary curve t(u), unlike the analogous fixed-background higher-dimensional expression.Rewriting the boundary condition in boundary time shows χr(u) transforms as a conformal primary of dimension 1−∆.
3.4 Coupling to matter: At leading order in GN, the boundary partition function is obtained by extremizing the total Schwarzian-plus-matter action, making the source dependence generally non-quadratic.The saddle t(u) generally depends on the boundary source χr(u).
3.4 Coupling to matter: For ∆ growing slower than G_N^-2/3 as GN→0, scalar backreaction is suppressed and the dual field is effectively free, with connected correlators beyond the two-point function vanishing.This follows because the Schwarzian action carries an extra 1/GN factor.
3.4 Coupling to matter: Leading gravitational corrections are computed by expanding the source-dependent exponential to quadratic order in Schwarzian fluctuations and contracting with their propagator.The one-point function of ε vanishes, so ⟨B(u1,u2)⟩ = 0.
3.4 Coupling to matter: The connected four-point correction depends significantly on Euclidean operator ordering because derivative contractions introduce sign functions of time differences.Alternating ordering produces cross distances u14 and u23 absent from the non-alternating result.

3.6 Relation to chaos

The notes connect semiclassical chaos to out-of-time-order correlators, whose early exponential growth is governed by a Lyapunov exponent and whose late-time saturation requires nonperturbative treatment.

Semiclassical chaos: In chaotic classical systems, nearby trajectories typically diverge exponentially after a perturbation to initial conditions.This dependence on initial conditions is the butterfly effect.
Quantum diagnostic: The quantum diagnostic squares a commutator and evaluates it thermally, producing four-point functions that include out-of-time-order correlators.The KMS relation identifies which terms have Lorentzian and out-of-time-order structure.
Quantum diagnostic: After a dissipation time t_d ∼ β, the commutator exhibits Lyapunov growth until the scrambling time t_s ∼ 1/λ_L log(1/ℏ), followed by saturation.The late-time saturation is called the Ruelle region.
Holographic interpretation: In holographic Einstein gravity, shockwaves show that Lyapunov growth results from exponential redshift near the black-hole horizon.The discussion identifies this behavior in the AdS-Schwarzschild setting.
Holographic interpretation: The out-of-time-order correlator grows through cross-distances, suggesting λ_L = 2π/β, the value also obtained for higher-dimensional black holes.This value is argued to be maximal for chaotic quantum systems with a classical limit.
Limitations: Perturbation theory in G_N cannot display the Ruelle region because neglected powers of G_N e^t become large near the scrambling time.Seeing saturation requires evaluating the Schwarzian path integral nonperturbatively in G_N.

4 SYK model

This section introduces the SYK model as an ensemble of finite-dimensional quantum-mechanical systems and identifies the references underlying the discussion.

4 SYK model: The section’s primary sources are references [24] [25], supplemented by seminar talks and additional references introduced throughout.The passage frames the section as a review of SYK-related material.

4.1 The model

The SYK model is a random finite-dimensional fermionic quantum system whose large-N and low-energy behavior makes it relevant to nearly-AdS2 physics and black-hole thermodynamics.

4.1 The model: SYK is an ensemble of finite-dimensional quantum-mechanical models whose Hamiltonians are finite Hermitian matrices built from gamma matrices.The gamma matrices furnish fermionic representations associated with the orthogonal group.
4.1 The model: The model uses all-to-all Gaussian couplings with variance scaling as 3!J/N^3/2, a scaling essential for interesting large-N behavior.A q-body generalization uses even q and admits an additional 1/q expansion.
4.1 Brief review: In the large-N limit, SYK classicalizes and is solvable through classical equations for two-dimensional master fields G and Σ.The model is defined using fermionic Clifford-algebra representations and random interactions.
4.1 Brief review: At low energies, SYK has emergent time-reparametrization symmetry spontaneously broken to SL(2,R), matching the Jackiw-Teitelboim symmetry-breaking pattern.The notes describe SYK as likely belonging to this infrared universality class despite lacking the considered holographic UV completions.
4.1 Brief review: For N = 20, the q = 2 spectrum has a long low-energy tail, whereas the q = 4 spectrum ends abruptly near random-matrix-like edges.The q = 2 model is quadratic and free; Gaussian-random-matrix spectral features are associated with quantum chaotic systems.
4.1 Brief review: The entropy formula shows noncommuting limits: finite-N SYK has few ground states, while the classical limit can exhibit macroscopic ground-state entropy.The notes interpret this as an extremal black-hole entropy without violating the third law.

4.2 Large N diagrammatics

Large-N diagrammatics reduce disorder-averaged SYK correlation functions to iterated melon diagrams and closed Schwinger-Dyson equations for the two-point function.

4.2 Large N diagrammatics: Perturbative SYK calculations average Feynman diagrams over Gaussian disorder using Wick’s theorem for the random couplings.The construction begins with a four-leg vertex for each realization.
4.2 Large N diagrammatics: The tadpole contribution vanishes because it is linear in J_ijkl, while melon diagrams provide the first nonzero contribution.The leading melon result does not scale with N.
4.2 Large N diagrammatics: Alternative disorder pairings are suppressed as N^-2 relative to the preceding nonzero diagrams.This distinguishes the dominant melon pairing from subleading contractions.
4.2 Large N diagrammatics: At leading order, only disorder pairings within a single melon survive without powers of N^-1 suppression, yielding an iterated structure.The self-energy Σ collects the iterated melon diagrams.
4.2 Large N diagrammatics: Resumming the diagrammatic series produces Schwinger-Dyson equations for the disorder-averaged two-point function.The equations are written using matrix multiplication notation for bilinear kernels.
4.2 Large N diagrammatics: For general q, the leading-order two-point function is determined by solving a closed set of integral equations whose second equation changes with q.The quadratic model instead has no large ground-state entropy and exhibits long spectral tails.

4.3 Master fields

The SYK disorder average is rewritten using collective fermion-bilinear and multiplier fields, producing an action proportional to N whose saddle equations reproduce the Schwinger-Dyson equations.

Disorder averaging: Averaging the partition function gives the annealed-disorder formulation, while averaging log Z uses replicas and is more physically relevant in many disordered systems.The two approaches agree to leading order in 1/N.
Collective fields: Introducing G and Σ makes G the fermion bilinear and Σ a Lagrange multiplier enforcing the corresponding delta constraint.The resulting exponent is bilinear in the fermions, enabling a Gaussian Berezin integral.
Large-N action: The Gaussian Berezin integral produces an effective action with an overall factor of N for the generalized q-body model.The scaling of the disorder variance is essential for obtaining this factor.
Saddle equations: Extremizing the collective-field action yields equations identical to the SYK Schwinger-Dyson equations.Thus the large-N dynamics is captured by a classical saddle-point problem.

4.4 Conformal limit

At frequencies much smaller than J, the SYK equations acquire an emergent reparametrization symmetry and admit conformal solutions, while the ultraviolet derivative term explicitly breaks that symmetry.

Solutions: The Schwinger-Dyson equations can be solved numerically, and the model is therefore solvable at large N.The analytic solution is known in the infrared and ultraviolet, with the intermediate regime τ ∼ J^-1 not known analytically.
Infrared symmetry: For frequencies ω ≪ J, dropping the derivative term produces infrared equations with an extra reparametrization symmetry.This approximation applies in the low-frequency regime relative to the interaction scale.
Conformal solutions: The collective fields transform as conformal two-point functions with conformal dimension Δ = 1/q.The conformal transformation leaves the infrared equations invariant.
Symmetry breaking: The emergent reparametrization symmetry is valid for |τ − τ′| ≫ J^-1 but is explicitly broken by the derivative term.The derivative term distinguishes the full equations from their infrared limit.
Symmetry breaking: The conformal solution spontaneously breaks reparametrization invariance down to SL(2, R), because Möbius transformations leave it unchanged.Finite-temperature correlators follow by reparametrizing the line solution to the thermal circle.

4.5 Large q and low temperature entropy

The large-q analysis gives the SYK thermal entropy and shows a nonzero zero-temperature limit, with a ground-state entropy that receives finite-q corrections and linear low-temperature behavior.

Motivation: The two limits relevant to the entropy do not commute, motivating evaluation of the large-N thermal free energy in a large-q expansion.The large-q expansion also simplifies other calculations in the model.
Caveats: The entropy claim is complicated by disorder averaging and by the formal q → ∞ limit, although melonic tensor models provide fixed-coupling examples without large global symmetries.For finite N, generic couplings are not expected to produce a large degeneracy.
Large-q calculation: The large-q saddle is obtained by parametrizing the correlator, solving its differential equation with thermal boundary conditions, and evaluating the saddle-point free energy.The dimensionless coupling βJ is controlled by a parameter v ranging from 0 to 1.
Thermal entropy: The exact-in-q large-N entropy has qualitatively similar behavior to the large-q result, and its zero-temperature limit is not zero.Figure 6 displays the thermal entropy as a function of T = (βJ)^-1 for q = 4, 6, 8, and ∞.
Low-temperature behavior: The low-temperature entropy contains a linear contribution in temperature, analogous to the Schwarzian result for near-extremal black holes.The correction π^2/(2q^2βJ) is identified as responsible for this behavior.
Ground-state entropy: The exact ground-state entropy is insensitive to the ultraviolet cutoff, whereas the ground-state energy depends on ultraviolet data.The formula vanishes at q = 2 and agrees with the β-independent part of the large-q expansion.

4.6 Schwarzian theory

The infrared reparametrization modes of SYK suggest a Schwarzian description, but deriving the nonlinear action requires care because the derivative expansion is ultraviolet divergent and non-systematic.

Reparametrization modes: The infrared saddle manifold is parametrized by reparametrizations because the derivative-free large-N action has emergent reparametrization symmetry.This creates a distinguished subset of fluctuation directions around the saddle.
Mode dynamics: The reparametrization action is suppressed by 1/J, making these modes easiest to excite when 1 ≪ βJ ≪ N and giving compatible thermodynamics.The suppression distinguishes them from the remaining path-integral directions.
Schwarzian proposal: The Schwarzian action is the unique lowest-derivative SL(2, R)-invariant candidate for the dynamics of these modes.The analogy with holographic theories motivates separating their dynamics in a strong-coupling expansion.
Ultraviolet problem: A formal derivative expansion is equivalent to a (βJ)^-1 expansion, but its terms are ultraviolet divergent despite the exact action being finite.At leading order, the expansion encounters an ill-defined δ(0)-type expression.
Regulated model: The modified SYK model introduces a cutoff ϵ = a0/J, preserving conformal solutions for τ ≫ ϵ while making the emergence of the Schwarzian action explicit.The regulated kernel smoothly connects ultraviolet and infrared behavior.
Limitations: The cutoff renders the 1/J expansion non-systematic, so higher-derivative terms may contribute at the same order and the nonlinear Schwarzian action is not established as a general SYK approximation.For q > 2, a different regulator or higher derivative order may be needed for a nonzero result.

4.7 Four point function

The four-point function is computed from fluctuations around the large-N SYK saddle, where its leading contribution is organized by ladder diagrams and analyzed through the inverse kernel. Conformal symmetry and contour deformation then expose the conformal primaries, scaling dimensions, and OPE data.

Kernel and fluctuations: The leading quantum correction to the fermion four-point function is obtained by studying small fluctuations around the large-N saddle.The fluctuation field is governed by a quadratic action whose kernel is Q, and the two-point function of the fluctuation is Q^-1.
Diagrammatic expansion: Each term in the expansion is a ladder diagram with n rungs, and these ladders give the leading 1/N diagrams for the four-point function.This is the four-point analogue of melonic dominance for the two-point function.
Conformal kernel: Conformal invariance makes the kernel diagonalizable using eigenfunctions of the SL(2,R) Casimir, reducing the problem to determining its eigenvalues.The relevant eigenfunctions of the Casimir are also eigenfunctions of the conformal kernel.
Spectrum and reparametrization modes: The discrete spectrum includes λ = 2n for n ∈ Z+, while the n = 1 term diverges because kc(2) = 1.This eigenvalue corresponds to reparametrization modes, whose conformal-limit zero modes receive (βJ)^-1 corrections described by the linearized Schwarzian theory.
Conformal-block decomposition: Contour deformation rewrites the four-point function as a conformal-block expansion whose poles determine the scaling dimensions and OPE coefficients of operators in the ψ × ψ OPE.The conformal blocks sum contributions from SL(2,R) descendants of each primary.
Operator content: For q = 4 and m ≫ 1, the operator dimensions follow the spectrum obtained from the pole equation, and the operators contain two ψ fields and 2m+1 derivatives plus an anomalous interaction contribution.The same propagating-mode spectrum can also be inferred directly from the quadratic fluctuation action.

4.8 Bulk dual?

The SYK bilocal fluctuations suggest an AdS2 description with infinitely many massive fields, but their nonlocal origin and kinematic-space interpretation make a naive local bulk dual inadequate. A redefinition can recover approximate massive AdS2 fields while retaining important limitations on locality and spectrum.

Bulk dual?: The large-N SYK degrees of freedom are bilocal fields G(τ1,τ2) and Σ(τ1,τ2), whose action and quadratic fluctuation action are nonlocal.Their dependence on two boundary coordinates makes a two-dimensional field interpretation tempting, but nonlocality obstructs a straightforward local bulk description.
Bulk dual?: A fluctuation profile peaked at an eigenvalue hm leads, after redefining φm = |τ12|gm, to an approximately massive scalar propagating on AdS2.The mass satisfies the d = 1 version of the usual AdS/CFT relation through the conformal weight.
Bulk dual?: The resulting bulk spectrum contains infinitely many fields with O(1) masses, unlike weakly coupled local gravity with finitely many light fields and parametrically heavy towers.The notes compare this spectrum to a stringy regime with ℓstring ∼ ℓAdS, while noting that the state count does not have Hagedorn growth.
Bulk dual?: Interpreting the bilocal space directly as holographic spacetime is too naive because its Laplacian is Lorentzian despite the Euclidean boundary theory.The bilocal is instead associated more naturally with kinematic space and geodesic operators, with local fields recovered through inverse X-ray transforms.

4.9 Outlook

The outlook surveys routes toward top-down, higher-dimensional, supersymmetric, tensor-model, and bulk-dual generalizations of SYK. These directions address ultraviolet completion, dimensionality, disorder, supersymmetry, and systematic bulk reconstruction, but often modify the original model’s infrared behavior.

Higher dimensional generalizations: In d ≥ 2 and q ≥ 4, fermion interactions become irrelevant or marginal, making strongly coupled SYK-like infrared physics difficult to obtain.The d = 2, q = 4 case is specifically marginal.
Higher dimensional generalizations: Higher-dimensional proposals use bosons or modified fermion kinetic terms to restore relevant interactions, with the latter preserving Lorentz invariance and melonic dominance.The modified-kinetic-term model flows to a conformal field theory in the infrared.
Supersymmetry: Supersymmetric generalizations construct an odd fermionic supercharge Q and Hamiltonian H = Q^2, yielding SYK-like physics with unbroken supersymmetry in the large-N ground state.The random coefficients defining the supercharge generate SYK-like interactions.
Tensor models: Tensor models remove the disorder average while reproducing the SYK large-N Schwinger-Dyson equations through additional contracted indices.Colored and uncolored tensor models can be represented as SYK Hamiltonians with structured zero-one couplings.
Bulk dual: Although SYK is neither a weakly coupled local gravitational theory nor a string theory, its large-N solvability may permit a systematic 1/N bulk Lagrangian with infinitely many weakly interacting fields.This bulk-dual program is identified with work pioneered in references [38] [48].
Towards top-down models: Top-down constructions seek stringy black-hole models that remain solvable at large N with SYK-like melonic dynamics.Certain matrix models already admit a double expansion in matrix size N and matrix number D, with melonic dominance.