AdS2/CFT1: JT gravity and the SYK model

These notes are a self-contained guide to the best-understood examples of $AdS_2/CFT_1$ holography:

• Jackiw–Teitelboim (JT) gravity, a universal effective theory of nearly $AdS_2$ regions (e.g. near-horizons of near-extremal black holes).
• The Sachdev–Ye–Kitaev (SYK) model, a solvable large-$N$ quantum mechanics whose IR dynamics is governed by the same universal Schwarzian mode.

A recurring lesson is that exact $AdS_2$ is “too rigid”. If you insist on the strict $AdS_2$ boundary conditions familiar from higher-dimensional AdS/CFT, then finite-energy excitations typically do not exist as normalizable states: backreaction forces you out of the strict $AdS_2$ asymptotics. The correct low-energy physics of near-extremal black holes is instead described by nearly $AdS_2$, where an approximate reparametrization symmetry is broken down to an exact $SL(2,\mathbb{R})$ subgroup and the surviving pseudo-Goldstone mode is described by the Schwarzian action. This same mode appears in SYK.

Conventions. We set $c=\hbar=k_B=1$. The $AdS_2$ radius is $L$ (often set to $L=1$ temporarily). The 2D Newton constant is $G$; in embeddings into higher dimensions, $G$ is related to the higher-dimensional $G_d$ by dimensional reduction.

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1. Why $AdS_2$ is special (and subtle)

1.1 The kinematics of $CFT_1$

A “$CFT_1$” is not a local QFT in the usual sense; it is conformal quantum mechanics. The global conformal group in 1D is the Möbius group acting on time:

$$SL(2,\mathbb{R}) :\quad \tau \mapsto \frac{a\tau+b}{c\tau+d},\qquad ad-bc=1.$$

It is generated by the familiar triplet

• $H$ (time translations),
• $D$ (dilatations),
• $K$ (special conformal transformations),

which can be represented as differential operators acting on functions of $\tau$:

$$H = \partial_\tau,\qquad D=\tau\,\partial_\tau,\qquad K=\tau^2\,\partial_\tau,$$

obeying the $sl(2,\mathbb{R})$ commutation relations

$$[D,H]=H,\qquad [D,K]=-K,\qquad [H,K]=2D.$$

A primary operator $\mathcal{O}(\tau)$ of scaling dimension $\Delta$ transforms under $SL(2,\mathbb{R})$ as

$$\mathcal{O}(\tau)\ \mapsto\ (c\tau+d)^{-2\Delta}\,\mathcal{O}\!\left(\frac{a\tau+b}{c\tau+d}\right).$$

Consequently, the $SL(2,\mathbb{R})$-invariant vacuum two-point function takes the usual conformal form

$$\langle \mathcal{O}(\tau)\mathcal{O}(0)\rangle \propto \frac{1}{|\tau|^{2\Delta}}.$$

Emergent reparametrizations and the Schwarzian

At large $N$, several quantum mechanical systems (SYK being the canonical example) exhibit an emergent (approximate) reparametrization symmetry

$$\tau \to f(\tau),$$

which is much larger than $SL(2,\mathbb{R})$. In an IR conformal regime, $SL(2,\mathbb{R})$ is typically exact, while the full $\mathrm{Diff}$ symmetry is:

• explicitly broken by UV effects (in SYK, by the $\partial_\tau$ term in the SD equations), and
• spontaneously broken by choosing a particular thermal state (or vacuum).

The corresponding pseudo-Goldstone mode is governed by the Schwarzian action, whose central object is the Schwarzian derivative

$$\{f(\tau),\tau\}\equiv \frac{f'''(\tau)}{f'(\tau)}-\frac{3}{2}\left(\frac{f''(\tau)}{f'(\tau)}\right)^2.$$

A crucial identity is that the Schwarzian is invariant under Möbius transformations:

$$f(\tau)\ \to\ \frac{a f(\tau)+b}{c f(\tau)+d}\quad\Rightarrow\quad \{f(\tau),\tau\}\ \text{is unchanged}.$$

This is why the physical configuration space of the Schwarzian mode is $\mathrm{Diff}/SL(2,\mathbb{R})$ rather than all of $\mathrm{Diff}$.

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1.2 The geometry of $AdS_2$ and its boundary

A convenient form of Poincaré $AdS_2$ is

$$ds^2=\frac{L^2}{z^2}\left(-dt^2+dz^2\right),\qquad z>0.$$

The boundary is at $z\to 0$, and in higher-dimensional holography one usually fixes the conformal class of the induced boundary metric there.

Another common coordinate system is global $AdS_2$:

$$ds^2 = L^2\left(-\cosh^2\rho\,dt^2 + d\rho^2\right), \qquad \rho\in(-\infty,\infty).$$

Equivalently, one can use

$$ds^2 = \frac{L^2}{\cos^2\sigma}\left(-d\tau^2 + d\sigma^2\right), \qquad \sigma\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right),$$

where the boundary components are at $\sigma\to \pm \pi/2$.

Key peculiarity: global $AdS_2$ has two disconnected boundaries (left and right). This already signals that $AdS_2$ holography is not a naïve dimensional reduction of $AdS_{d\ge 3}$ intuition.

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1.3 Pure 2D Einstein gravity is trivial

In 2D, the Einstein tensor vanishes identically:

$$G_{\mu\nu}\equiv R_{\mu\nu}-\frac12 R\,g_{\mu\nu}=0.$$

Equivalently, the Einstein–Hilbert action in 2D computes a topological invariant (the Euler characteristic):

$$\frac{1}{16\pi G}\int_M d^2x\,\sqrt{g}\,R \propto \chi(M).$$

So pure 2D Einstein gravity has no local dynamics: there are no propagating graviton degrees of freedom.

To obtain a dynamical theory that still captures near-horizon physics, one introduces a dilaton field $\Phi$. In many higher-dimensional settings, $\Phi$ arises from dimensional reduction and measures the transverse area of the compact space (e.g. an $S^{d-2}$):

$$\Phi\ \sim\ \frac{\mathrm{Area}(\mathcal{K})}{4G_d}.$$

The universal nearly-$AdS_2$ effective theory of this type is JT gravity.

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1.4 Backreaction in $AdS_2$: why “nearly $AdS_2$” is the right concept

A useful way to summarize the subtlety is:

• In higher-dimensional AdS/CFT, one usually fixes the boundary metric (up to Weyl rescaling) and studies normalizable bulk excitations.
• In strict $AdS_2$, with the most naïve analog of those boundary conditions, finite-energy excitations typically backreact so strongly that the would-be $AdS_2$ boundary conditions are destroyed.

In near-extremal black holes, the correct low-energy sector is therefore not an autonomous “$CFT_1$ with a standard stress tensor,” but rather a nearly-$AdS_2$ throat whose only universal degree of freedom is a boundary reparametrization. This is precisely the Schwarzian mode.

A concrete way to see this in JT gravity is that the bulk curvature is fixed ($R=-2/L^2$), so the bulk metric is locally $AdS_2$ and the only dynamical imprint of energy above extremality is encoded in the boundary curve (or equivalently, the reparametrization function).

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2. Nearly $AdS_2$ from near-extremal black holes

2.1 Near-horizon $AdS_2\times \mathcal{K}$

Many extremal black holes (charged, rotating, supersymmetric, etc.) have a near-horizon throat containing an $AdS_2$ factor,

$$ds^2 \approx ds^2_{AdS_2} + ds^2_{\mathcal{K}},$$

where $\mathcal{K}$ is a compact space (often an $S^{d-2}$).

A standard example is the extremal Reissner–Nordström black hole in 4D, for which the near-horizon limit yields

$$ds^2 \approx L_2^2\left(-\rho^2 dt^2 + \frac{d\rho^2}{\rho^2}\right) + r_0^2\,d\Omega_2^2,$$

i.e. $AdS_2(L_2)\times S^2(r_0)$. (The specific relation between $L_2$ and $r_0$ depends on the UV completion and conventions; the essential point is the appearance of an $AdS_2$ throat.)

For a near-extremal black hole, the throat is not exactly $AdS_2$: it is a long, nearly-$AdS_2$ region whose departure from $AdS_2$ is controlled by a small energy above extremality. The universal low-energy description of this departure is JT gravity.

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2.2 Dimensional reduction and the dilaton

Under reduction from $d$ dimensions to 2 dimensions, one typically writes an ansatz of the schematic form

$$ds_d^2 = g_{\mu\nu}(x)\,dx^\mu dx^\nu + \cdots,$$

where the omitted terms include a compact metric with an overall scale factor depending on the 2D coordinates. In the simplest (spherically symmetric) case,

$$ds_d^2 = g_{\mu\nu}(x)\,dx^\mu dx^\nu + r(x)^2\,d\Omega_{d-2}^2,$$

and the dilaton is essentially the transverse area,

$$\Phi(x)\ \propto\ r(x)^{d-2}.$$

Fluctuations of $\Phi$ are not optional: they encode how the near-horizon throat glues to the asymptotic region. In the nearly-$AdS_2$ regime, the metric remains locally $AdS_2$ while the dilaton becomes dynamical and carries the information about energy and entropy above extremality.

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3. Jackiw–Teitelboim (JT) gravity

JT gravity is the simplest 2D dilaton gravity that enforces constant negative curvature and captures universal near-extremal dynamics.

3.1 Action and equations of motion

A standard Euclidean JT action is

$$I_{\mathrm{JT}} = -\frac{1}{16\pi G}\left[\int_M d^2x\,\sqrt{g}\,\Phi\left(R+\frac{2}{L^2}\right) +2\int_{\partial M} du\,\sqrt{h}\,\Phi\,K\right] + I_{\mathrm{ct}}.$$

Here

• $h$ is the induced metric on the boundary curve $\partial M$,
• $K$ is the extrinsic curvature (in 1D: the geodesic curvature) of that curve,
• the boundary term is the 2D analog of the Gibbons–Hawking–York term ensuring a good variational principle for Dirichlet boundary conditions on the metric, and
• $I_{\mathrm{ct}}$ is a counterterm that removes divergences at the asymptotic boundary and implements holographic renormalization.

A convenient counterterm is

$$I_{\mathrm{ct}}=+\frac{1}{8\pi G}\int_{\partial M} du\,\sqrt{h}\,\frac{\Phi}{L}.$$

Equivalently, one often packages the boundary terms as

$$I_{\mathrm{bdy}}=-\frac{1}{8\pi G}\int_{\partial M} du\,\sqrt{h}\,\Phi\left(K-\frac{1}{L}\right),$$

which makes it clear that only the deviation of $K$ from its $AdS_2$ value contributes after renormalization.

Topological term and extremal entropy

It is often useful to split the dilaton into a constant part and a fluctuating part,

$$\Phi = \Phi_0 + \phi.$$

The $\Phi_0$ piece multiplies the Euler characteristic:

$$-\frac{1}{16\pi G}\int_M \sqrt{g}\,\Phi_0 R -\frac{1}{8\pi G}\int_{\partial M}\sqrt{h}\,\Phi_0 K = -\frac{\Phi_0}{4G}\,\chi(M).$$

This produces a constant contribution to the black hole entropy,

$$S_0=\frac{\Phi_0}{4G},$$

interpreted as the extremal (zero-temperature) entropy.

Equations of motion

Varying $I_{\mathrm{JT}}$ with respect to $\Phi$ gives the constant-curvature condition

$$R=-\frac{2}{L^2}.$$

So the metric is locally $AdS_2$.

Varying with respect to $g_{\mu\nu}$ gives a linear equation for the dilaton:

$$\nabla_\mu\nabla_\nu \Phi - g_{\mu\nu}\nabla^2\Phi + \frac{1}{L^2}g_{\mu\nu}\Phi = 0$$

in vacuum. With matter coupled to the 2D metric, the right-hand side becomes proportional to the matter stress tensor (details depend on the matter coupling).

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3.2 Classical solutions: $AdS_2$ and the JT black hole

Because $R$ is fixed, solutions are locally $AdS_2$ but can differ globally and through the dilaton profile.

A convenient Lorentzian “black hole patch” of $AdS_2$ is

$$ds^2 = -\frac{r^2-r_h^2}{L^2}\,dt^2 + \frac{L^2}{r^2-r_h^2}\,dr^2,$$

with a dilaton profile

$$\Phi(r)=\Phi_0 + \Phi_r\,\frac{r}{L}.$$

Here $\Phi_r$ sets the slope of the dilaton and is fixed by the UV completion (e.g. by matching to the higher-dimensional geometry).

The Hawking temperature is

$$T=\frac{r_h}{2\pi L^2},$$

and the entropy is determined by the dilaton at the horizon:

$$S=\frac{\Phi(r_h)}{4G}=S_0+\frac{\Phi_r}{4G}\frac{r_h}{L}.$$

Eliminating $r_h$ in favor of $T$ yields the characteristic linear-in-$T$ near-extremal entropy:

$$S(T)=S_0 + 2\pi \left(\frac{\Phi_r L}{4G}\right) T.$$

The combination in parentheses is (up to conventions) the same parameter that becomes the Schwarzian coupling.

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3.3 Boundary conditions and the boundary reparametrization mode

To define “nearly $AdS_2$” precisely, one introduces a cutoff boundary curve close to the asymptotic region and fixes two pieces of data:

1. The proper boundary line element (i.e. the induced metric along the curve).
2. The leading divergence of the dilaton along the curve.

A standard choice is to require the induced metric to be

$$ds^2_{\partial} = \frac{du^2}{\epsilon^2},$$

where $u$ is the boundary time and $\epsilon$ is a UV cutoff. One also fixes

$$\Phi\big|_{\partial} = \frac{\phi_r}{\epsilon},$$

where $\phi_r$ is a renormalized parameter kept fixed as $\epsilon\to 0$.

The boundary curve in Poincaré $AdS_2$

Work in Euclidean Poincaré $AdS_2$,

$$ds^2=\frac{L^2}{z^2}\left(dt^2+dz^2\right).$$

A boundary curve is described by an embedding $(t(u),z(u))$. Imposing $ds^2_{\partial}=du^2/\epsilon^2$ fixes the curve to be asymptotically

$$t(u)=f(u),\qquad z(u)=\epsilon\,f'(u)+O(\epsilon^3),$$

for some monotonic function $f(u)$. So the only remaining dynamical degree of freedom is the reparametrization $f(u)$: it tells you how the physical boundary time $u$ is embedded into the “natural” $AdS_2$ time $t$.

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3.4 The Schwarzian action from JT: the essential derivation

On-shell, the bulk term vanishes because $R=-2/L^2$, so the effective action reduces to the renormalized boundary term:

$$I_{\mathrm{JT,on\text{-}shell}} = -\frac{1}{8\pi G}\int_{\partial M} du\,\sqrt{h}\,\Phi\left(K-\frac{1}{L}\right) - S_0\,\chi(M).$$

Now evaluate $K$ for the near-boundary curve. One finds the universal expansion

$$K = \frac{1}{L}\left(1 + \epsilon^2\,\{f(u),u\} + O(\epsilon^4)\right).$$

Plugging $\sqrt{h}=1/\epsilon$ and $\Phi=\phi_r/\epsilon$ into the boundary action gives

$$I_{\mathrm{JT,on\text{-}shell}} = -S_0 - C\int du\,\{f(u),u\} + O(\epsilon^2),$$

where the Schwarzian coupling is

$$C = \frac{\phi_r}{8\pi G}.$$

(Depending on conventions and how $L$ is scaled, you may encounter $C=\phi_r L/(8\pi G)$; what matters is that $C$ has dimensions of energy$\times$time$^2$ and controls the near-extremal specific heat.)

Two structural points are worth emphasizing:

1. Symmetry: the Schwarzian is invariant under $SL(2,\mathbb{R})$, so the physical configuration space is

   $$\mathrm{Diff}(S^1)/SL(2,\mathbb{R}).$$

2. Universality: any UV completion producing a long nearly-$AdS_2$ throat yields the same low-energy action; all model dependence is encoded in $S_0$ and $C$.

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3.5 Thermodynamics from the Schwarzian

At finite temperature $\beta^{-1}$ we work on the Euclidean circle $u\sim u+\beta$. The classical saddle corresponding to a smooth Euclidean black hole is represented by a map with constant Schwarzian, for example

$$f(u)=\tan\left(\frac{\pi u}{\beta}\right) \quad\Rightarrow\quad \{f(u),u\}=\frac{2\pi^2}{\beta^2}.$$

Plugging this into the Schwarzian action yields the leading low-temperature partition function

$$\log Z(\beta)=S_0+\frac{2\pi^2 C}{\beta}+\cdots.$$

Hence

$$E(\beta)=-\partial_\beta \log Z = \frac{2\pi^2 C}{\beta^2}+\cdots, \qquad S(\beta)=\log Z+\beta E = S_0+\frac{4\pi^2 C}{\beta}+\cdots.$$

This reproduces the hallmark near-extremal scaling:

$$E\propto T^2,\qquad S-S_0\propto T.$$

Beginner note: the exact Schwarzian path integral can be solved (see [11]) and yields a universal density of states with a characteristic $\sinh$ behavior at high energy. The precise prefactors depend on the normalization convention for $C$, so it is best to consult a reference when you need the exact expression.

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3.6 Matter correlators in nearly $AdS_2$

A simple way to include probe matter is to consider a bulk scalar of mass $m$, dual to a boundary primary operator $\mathcal{O}(u)$ of dimension $\Delta$ related by

$$m^2 L^2 = \Delta(\Delta-1).$$

For a fixed reparametrization $f(u)$, the conformal two-point function takes the universal form

$$\langle \mathcal{O}(u_1)\mathcal{O}(u_2)\rangle_f \propto \left( \frac{f'(u_1)f'(u_2)}{\left[f(u_1)-f(u_2)\right]^2} \right)^{\Delta}.$$

The physical correlator is obtained by integrating over $f$ with weight $e^{-I_{\mathrm{Sch}}[f]}$. This “gravitational dressing by the Schwarzian” is the robust universal prediction of nearly-$AdS_2$ gravity.

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3.7 Maximal chaos and shockwaves

Nearly-$AdS_2$ gravity exhibits maximal Lyapunov growth in out-of-time-order correlators (OTOCs). The universal statement is

$$\lambda_L = \frac{2\pi}{\beta},$$

which saturates the general chaos bound. On the gravity side, this is tied to high-energy scattering near the horizon (shockwave physics). In the Schwarzian theory, it comes from the dynamics of the reparametrization mode, which enhances certain four-point functions and produces the exponential growth.

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4. The SYK model

SYK is a solvable large-$N$ quantum mechanics of $N$ Majorana fermions with random all-to-all interactions. It provides a concrete microscopic realization of the Schwarzian mode and of maximal chaos.

4.1 Definition

The (Majorana) SYK Hamiltonian with even $q$-fermion interactions is

$$H = i^{q/2}\sum_{1\le i_1<\cdots<i_q\le N} J_{i_1\cdots i_q}\,\psi_{i_1}\cdots\psi_{i_q},$$

with Majorana fermions obeying

$$\{\psi_i,\psi_j\}=\delta_{ij}.$$

The couplings are drawn from a Gaussian ensemble with

$$\langle J_{i_1\cdots i_q}\rangle=0, \qquad \langle J_{i_1\cdots i_q}^2\rangle = \frac{(q-1)!\,J^2}{N^{q-1}}.$$

The scaling with $N$ is chosen so that the model has a sensible large-$N$ limit: typical energies remain $O(N)$ and correlation functions admit a controlled $1/N$ expansion.

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4.2 Disorder average and bilocal formulation

After averaging over $J_{i_1\cdots i_q}$ (using replicas, supersymmetry, or self-averaging assumptions), the theory can be rewritten in terms of bilocal collective fields:

• the fermion two-point function $G(\tau_1,\tau_2)$, and
• a self-energy $\Sigma(\tau_1,\tau_2)$ enforcing the definition of $G$.

At large $N$, the effective action takes the schematic form

$$\frac{I_{\mathrm{eff}}[G,\Sigma]}{N} = -\frac12 \log\det(\partial_\tau-\Sigma) +\frac12 \int d\tau_1 d\tau_2\left[\Sigma\,G-\frac{J^2}{q}G^q\right].$$

Varying with respect to $G$ and $\Sigma$ gives the Schwinger–Dyson equations. In frequency space,

$$G(i\omega_n)^{-1}=-i\omega_n-\Sigma(i\omega_n), \qquad \Sigma(\tau)=J^2\,G(\tau)^{q-1},$$

and in the time domain one often writes the first equation as a convolution:

$$(\partial_{\tau_1}-\Sigma)\circ G = \delta,$$

meaning

$$\int d\tau_3\,(\partial_{\tau_1}\delta(\tau_1-\tau_3)-\Sigma(\tau_1,\tau_3))\,G(\tau_3,\tau_2)=\delta(\tau_1-\tau_2).$$

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4.3 Emergent conformal regime and reparametrizations

In the deep IR, $|\omega|\ll J$ (equivalently $\beta J\gg 1$ at finite temperature), one can often neglect the $-i\omega_n$ term. The SD equations then become approximately invariant under time reparametrizations

$$\tau \to f(\tau),$$

with the transformation law

$$G(\tau_1,\tau_2)\to \left[f'(\tau_1)f'(\tau_2)\right]^{\Delta}\,G(f(\tau_1),f(\tau_2)), \qquad \Delta=\frac{1}{q}.$$

A conformal solution at zero temperature is

$$G_c(\tau)=b\,\frac{\mathrm{sgn}(\tau)}{|J\tau|^{2\Delta}}, \qquad \Delta=\frac{1}{q},$$

with $b$ fixed by the SD equations. A common (convention-dependent) expression is

$$b^q=\frac{1}{2\pi}(1-2\Delta)\tan(\pi\Delta).$$

At finite temperature, the conformal correlator is obtained by mapping the line to the circle:

$$G_c(\tau)=b\left(\frac{\pi/\beta}{J\sin(\pi\tau/\beta)}\right)^{2\Delta}\mathrm{sgn}(\tau).$$

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4.4 Explicit breaking and the Schwarzian action in SYK

The UV term $-i\omega_n$ (or $\partial_\tau$ in the time domain) explicitly breaks reparametrization symmetry. The resulting low-energy effective action for the soft mode is again the Schwarzian:

$$I_{\mathrm{Sch}}[f] = -C_{\mathrm{SYK}}\int_0^\beta d\tau\,\{f(\tau),\tau\},$$

with

$$C_{\mathrm{SYK}} = \alpha_S(q)\,\frac{N}{J},$$

where $\alpha_S(q)$ is a known positive $q$-dependent number (see e.g. [6,10]).

This is one of the sharpest pieces of evidence for the nearly-$AdS_2$ interpretation: the same effective theory controls both SYK and JT gravity.

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4.5 Four-point function, ladder diagrams, and chaos

The connected four-point function at large $N$ is dominated by ladder diagrams. One introduces a kernel $\mathcal{K}$ acting on bilocal functions; schematically,

$$\mathcal{F} = \mathcal{F}_0 + \mathcal{K}\circ \mathcal{F},$$

so $\mathcal{F}$ is controlled by eigenvalues of $\mathcal{K}$. A crucial role is played by an eigenmode corresponding to an operator of dimension $h=2$, associated to reparametrizations. This is precisely the mode captured by the Schwarzian effective action.

In real time, the OTOC exhibits exponential growth controlled by the Lyapunov exponent

$$\lambda_L=\frac{2\pi}{\beta},$$

saturating the chaos bound. The associated scrambling time scales as

$$t_* \sim \frac{\beta}{2\pi}\log N,$$

up to $q$-dependent factors and corrections at finite coupling.

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4.6 Spectral statistics and random-matrix behavior

At very late times, SYK exhibits spectral correlations well described by random matrix theory, including the “ramp” and “plateau” in spectral form factors. This is closely related to the fact that Euclidean JT gravity, when defined by summing over topologies, computes ensemble-averaged quantities—see Section 5.4.

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5. The JT/SYK correspondence: what matches and what it means

5.1 The universal low-energy sector

A clean and widely used statement is not “JT is dual to SYK” in precisely the same sense as higher-dimensional AdS/CFT, but rather:

JT gravity captures the universal, symmetry-controlled low-energy sector of near-extremal black holes, and SYK provides an explicit solvable quantum system whose low-energy sector is governed by the same Schwarzian dynamics.

The matching includes:

• Thermodynamics: $S-S_0\propto T$ and $E\propto T^2$ with the same Schwarzian coefficient.
• Correlation functions: the conformal form dressed by reparametrizations matches.
• Chaos: $\lambda_L=2\pi/\beta$ and the structure of OTOCs match.
• Operator structure: the reparametrization (stress-tensor-like) sector in SYK corresponds to boundary graviton/dilaton fluctuations in JT.

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5.2 A practical dictionary of parameters

There are two key parameters on the JT side:

• $S_0$ (or $\Phi_0$): the extremal entropy.
• $C$ (or $\phi_r/G$): the Schwarzian coupling controlling the near-extremal specific heat.

On the SYK side, at large $N$ and strong coupling ($\beta J\gg 1$):

• $S_0$ is the residual entropy at $T\to 0$ (large at large $N$).
• $C_{\mathrm{SYK}}=\alpha_S(q)N/J$ controls the Schwarzian effective action.

Schematically,

$$S_0^{\mathrm{JT}} \leftrightarrow S_0^{\mathrm{SYK}}, \qquad C^{\mathrm{JT}} \leftrightarrow C^{\mathrm{SYK}}.$$

Once this identification is made, many low-energy observables coincide.

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5.3 What should “$CFT_1$” mean here?

A common source of confusion is what “$CFT_1$” should mean in $AdS_2/CFT_1$. Three useful viewpoints are:

1. Effective-sector viewpoint: near-extremal black holes have a universal nearly-$AdS_2$ sector described by the Schwarzian; SYK realizes the same sector microscopically.

2. Ensemble viewpoint (Euclidean JT): when JT is defined by summing over Euclidean topologies, the path integral computes an ensemble average over boundary quantum systems. This is powerful for reproducing spectral correlations, but it means the “dual” is not a single fixed Hamiltonian.

3. Embedding viewpoint (UV completion): in string theory / higher-dimensional gravity, an $AdS_2$ throat is embedded into a UV-complete theory with additional degrees of freedom. The full dual is a standard CFT (or quantum system) in the UV, and the $AdS_2$ dynamics describes a subsector.

Which viewpoint is appropriate depends on the question you are asking. For many low-energy thermodynamic and chaotic observables, the Schwarzian sector is the robust universal content.

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5.4 Wormholes, factorization, and matrix integrals (important caveat)

If one sums over Euclidean topologies in JT gravity, one finds contributions from wormholes connecting multiple boundaries. This reproduces universal random-matrix correlations and leads naturally to a matrix-integral description of JT gravity.

However, wormhole contributions imply that multi-boundary partition functions do not factorize in the naïve way expected of a single quantum theory:

$$Z(\beta_1,\beta_2)\neq Z(\beta_1)\,Z(\beta_2)$$

in the gravitational computation. The modern understanding is that the JT path integral computes ensemble-averaged quantities, for which such non-factorization is expected.

In higher-dimensional black holes, the nearly-$AdS_2$ region is a low-energy effective description inside a larger UV-complete theory, and one should be careful about which “sum over topologies” is physically appropriate.

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6. Beyond the minimal setup (brief roadmap)

Standard directions beyond the simplest JT/SYK story include:

• Complex SYK and $U(1)$ charge: charged nearly-$AdS_2$ gravity and additional soft modes (phase mode + Schwarzian).
• Supersymmetric SYK: additional structure and protected sectors.
• Tensor models (no disorder): SYK-like solvability without quenched randomness.
• Extra matter in $AdS_2$: modifies operator content while retaining the universal Schwarzian sector at low energies.
• Embedding into string theory: relating $S_0$ and $C$ to microscopic degeneracies and moduli of higher-dimensional black holes.

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[15] E. Witten, An SYK-Like Model Without Disorder, J. Phys. A 52, 474002 (2019) [arXiv:1610.09758].

[16] I. R. Klebanov and G. Tarnopolsky, Uncolored Random Tensors, Melon Diagrams, and the SYK Models, Phys. Rev. D 95, 046004 (2017) [arXiv:1611.08915].

[17] D. Grumiller, W. Kummer and D. V. Vassilevich, Dilaton Gravity in Two Dimensions, Phys. Rept. 369, 327 (2002) [arXiv:hep-th/0204253].

[18] A. Kitaev, A Simple Model of Quantum Holography, KITP program “Entangled Quantum Matter” (2015). (See also the early SYK literature and later write-ups cited above.)