Near-AdS2, SYK, and the Duality Interface

Near-AdS₂ gravity and the Sachdev–Ye–Kitaev (SYK) model share an infrared reparametrization mode, Schwarzian thermodynamics, conformal two-point scaling, and a maximal leading chaos window. Those matches define an effective duality interface. They do not equate ultraviolet spectra, global symmetries, disorder averaging, or complete Hilbert spaces.

Required background. SYK Models and Local Quantum Criticality supplies the many-body model. JT Gravity and the Schwarzian Boundary Mode supplies the gravitational theory.

Helpful background. Holographic Duality: Claims, Dictionaries, and Regimes supplies the claim structure. Lyapunov Growth and Chaos Bounds supplies the chaos interpretation.

Evidence cutoff: 25 July 2026.

The two low-energy descriptions

For even interaction order $q$, a Majorana SYK Hamiltonian is

$$H=i^{q/2}\sum_{i_1<\cdots<i_q} J_{i_1\cdots i_q}\chi_{i_1}\cdots\chi_{i_q},$$

with Gaussian couplings normalized by

$$\overline{J_{i_1\cdots i_q}^2} =\frac{(q-1)!J^2}{N^{q-1}}.$$

At $N\gg1$ and $\beta J\gg1$,

$$\log Z_{\mathrm{SYK}} =Ns_0+\frac{2\pi^2C_{\mathrm{SYK}}}{\beta} +O((\beta J)^{-2}), \qquad C_{\mathrm{SYK}}=\frac{N\alpha_S(q)}{J}.$$

JT gravity gives the same functional form,

$$\log Z_{\mathrm{JT}} =S_0+\frac{2\pi^2C_{\mathrm{JT}}}{\beta}+\cdots .$$

The effective matching is therefore

$$S_0\leftrightarrow Ns_0, \qquad C_{\mathrm{JT}}\leftrightarrow\frac{N\alpha_S(q)}{J}.$$

It is a map between infrared parameters, not an equality of microscopic Lagrangians.

Entropy and heat capacity as the first application

Differentiate both partition functions:

$$S(T)=S_0+4\pi^2CT+\cdots, \qquad C_V=4\pi^2CT+\cdots .$$

Matching the entropy slope fixes $C$; the heat capacity is then a prediction rather than a second fit. Bilocal correlators provide another test: the SYK fermion has $\Delta=1/q$, and its reparametrized conformal two-point function has the same Schwarzian dressing as a JT bilocal of dimension $h=\Delta$. The low-energy SYK solution and its four-point kernel were derived by Maldacena and Stanford 2016.

Both theories also give $\lambda_L=2\pi/\beta$ at leading strong coupling, but this equality follows from the shared soft mode. It is not independent evidence for the ultraviolet dictionary.

What does not match automatically

Structure	SYK input	JT input	Status
ultraviolet levels	finite-$N$ random Hamiltonian	absent from the Schwarzian EFT	unmatched
global symmetry	chosen fermion and coupling symmetry class	chosen matter and boundary conditions	model-dependent
averaging	quenched or annealed disorder prescription	topology or matrix-ensemble interpretation	requires an additional claim
high-frequency correlator	set by $J$ and microscopic kernel	set by parent throat completion	unmatched
nonperturbative spectrum	one realization or disorder ensemble	completion-dependent	not fixed by the IR map

The general near-AdS₂/SYK connection was sharpened through their shared symmetry-breaking pattern Jensen 2016.

Ultraviolet adversary

Choose two SYK-like Hamiltonians with the same $s_0$ and $C$ but different symmetry class or ultraviolet spectral density. Their leading Schwarzian thermodynamics agrees, while high-frequency correlators and finite-$N$ level correlations differ. A unique microscopic dual cannot be inferred from the common infrared coefficients.

The defensible conclusion is an infrared effective correspondence over $1/N\ll\omega/J,T/J\ll1$, refined according to the observable. Extending it to exact spectra, a single realization, or a unique quantum gravity requires new information not contained in the Schwarzian match.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Jensen, Kristan. “Chaos in AdS₂ Holography.” Physical Review Letters 117, 111601 (2016). DOI.
• Maldacena, Juan, and Douglas Stanford. “Remarks on the Sachdev–Ye–Kitaev Model.” Physical Review D 94, 106002 (2016). DOI.