Near-AdS2, SYK, and the Duality Interface

Near-AdS₂ gravity and the Sachdev–Ye–Kitaev (SYK) model can flow to the same low-energy theory of a reparametrization mode. The central quantitative statement is a match of one effective coupling, the Schwarzian coefficient $C$. Once conventions are fixed, that match relates the linear-in-temperature entropy, the heat capacity, the soft dressing of suitable correlators, and the leading strong-coupling chaos response. It does not by itself identify the microscopic Hilbert spaces, the zero-temperature entropy, the complete operator spectra, or a unique nonperturbative theory of quantum gravity.

Here a duality interface means a controlled dictionary between two routes to this shared infrared effective theory. It is not a spacetime interface or defect, a third theory inserted between SYK and gravity, or an exact equivalence of complete finite-$N$ theories. The purpose of the interface is to make every comparison answer four questions: which observable is being matched, in which regime, with which normalization, and with which corrections.

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: 28 August 2026.

The interface is a shared infrared effective theory

The two constructions reach the reparametrization mode by different routes.

• In nearly AdS₂ gravity, the asymptotic boundary conditions leave a fluctuating embedding of the regulated boundary curve. Its coordinate $f(u)$ is a gravitational boundary degree of freedom.
• In SYK, the large-$N$ collective fields have an approximate reparametrization symmetry in the conformal limit. Acting with $f(\tau)$ deforms the bilocal saddle without changing the conformal equations.

In both cases the exact $PSL(2,\mathbb R)$ subgroup is redundant, while the larger reparametrization symmetry is explicitly broken: by the nearly AdS₂ dilaton boundary condition on the gravity side and by the microscopic fermion kinetic term on the SYK side. The resulting pseudo-Goldstone mode lives on

$$\frac{\operatorname{Diff}^{+}(S^1)}{PSL(2,\mathbb R)}.$$

“Pseudo-Goldstone” is useful language here: the mode would be exactly flat in the ideal conformal limit, but the small explicit breaking gives it a nonzero action. Matching the two actions identifies the dynamics of this soft coordinate. It does not identify the JT boundary curve with an SYK fermion, nor does it reconstruct either microscopic theory from the other. The symmetry-breaking argument and its nearly AdS₂ realization are developed in Maldacena, Stanford, and Yang 2016, §§ 3.1 and 4–6.

Two routes to the Schwarzian mode

We use $\hbar=k_B=1$ and Euclidean time $u\sim u+\beta$, with $\beta=1/T$. On the SYK side, take $N$ Majorana fermions with

$$\{\chi_i,\chi_j\}=\delta_{ij},$$

and fixed even interaction order $q\geq4$, with $q\ll N$:

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

The subscript $J$ labels one coupling realization. For the standard Gaussian ensemble,

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

The overbar denotes the disorder average. The restriction $q\geq4$ excludes the quadratic, integrable $q=2$ model. This normalization, together with $\{\chi_i,\chi_j\}=\delta_{ij}$, fixes what is meant below by $J$ and by the dimensionless coefficient $\alpha_S(q)$. Changing either convention changes the numerical value assigned to $\alpha_S$.

Maldacena and Stanford use both the ordinary coupling $J$ in the displayed variance and the rescaled interaction

$$\mathcal J=\frac{\sqrt q}{2^{(q-1)/2}}J.$$

Their Schwarzian coefficient $\alpha_S(q)$ multiplies $N/J$. If instead one chooses to write $C_{\mathrm{SYK}}=N\widehat\alpha_S(q)/\mathcal J$, the converted coefficient is

$$\widehat\alpha_S(q) =\frac{\sqrt q}{2^{(q-1)/2}}\, \alpha_S(q), \qquad \frac{N\alpha_S(q)}{J} =\frac{N\widehat\alpha_S(q)}{\mathcal J}.$$

Thus $J$ is the coupling in the displayed variance and in the coefficient match below, whereas $\mathcal J$ is convenient for strong-coupling and large-$q$ formulas. Using the reciprocal conversion would fail the $q=4$ checkpoint below.

For a monotone thermal reparametrization satisfying $f(u+\beta)=f(u)+\beta$, define

$$\{g,u\} =\frac{g'''(u)}{g'(u)} -\frac{3}{2}\left(\frac{g''(u)}{g'(u)}\right)^2.$$

The two leading Euclidean actions can then be written in the same convention:

$$\begin{aligned} I_{\mathrm{JT}}[f] &=-C_{\mathrm{JT}}\int_0^\beta du\, \left\{\tan\frac{\pi f(u)}{\beta},u\right\}, & C_{\mathrm{JT}}&=\frac{\phi_r}{8\pi G_2},\\ I_{\mathrm{SYK}}[f] &=-C_{\mathrm{SYK}}\int_0^\beta du\, \left\{\tan\frac{\pi f(u)}{\beta},u\right\} +\delta I_{\mathrm{UV}}[f], & C_{\mathrm{SYK}}&=\frac{N\alpha_S(q)}{J}. \end{aligned}$$

Here $\phi_r$ is the renormalized boundary dilaton in the convention of the JT prerequisite, $G_2$ is the two-dimensional Newton constant, and $\delta I_{\mathrm{UV}}$ collects finite-coupling and higher-derivative corrections. The SYK coefficient follows from the lifting of the reparametrization zero mode Maldacena and Stanford 2016, § 4, Eqs. (4.176)–(4.179); the JT coefficient and thermal saddle follow from Maldacena, Stanford, and Yang 2016, § 3.1, Eqs. (3.18)–(3.21).

The reproducible soft-sector match is therefore

$$\boxed{ C_{\mathrm{JT}} =\frac{\phi_r}{8\pi G_2} =\frac{N\alpha_S(q)}{J} =C_{\mathrm{SYK}} }.$$

Every equality in this box depends on the displayed normalizations. A fast dimensional check is that $C$ has units of inverse energy, so $C/\beta$ and $CT$ are dimensionless. A deeper check is thermodynamic: the same $C$ must independently reproduce the entropy slope, the energy above extremality, and the heat capacity.

For $q=4$, a useful numerical round-trip in this convention is

$$4\pi^2 C_{\mathrm{SYK}} \simeq 0.396\,\frac{N}{J}, \qquad C_{\mathrm{SYK}} \simeq 0.0100\,\frac{N}{J}.$$

This checkpoint, reported in Maldacena and Stanford 2016, § 5, Eqs. (5.181)–(5.183), makes a hidden factor of $2$, $\pi$, or $\mathcal J/J$ immediately visible.

The full nonlinear SYK action is an effective-field-theory statement, not merely the quadratic soft eigenvalue of a four-point kernel. Kitaev and Suh organize its symmetry constraints and leading nonlocal correction in Kitaev and Suh 2018, § 2.2, Eqs. (14)–(18). A later microscopic derivation at controlled large interaction order strengthens that statement in its own limit, but does not turn the match into a proof for every finite $q$ or every SYK-like model Bucca and Mezei 2025, §§ 2–4.

The overlap is a hierarchy, not one inequality

Three low-energy regimes must be separated. Suppressing coefficients of order unity, the relevant scales obey

$$\mathcal J\gg C^{-1}\sim\frac{\mathcal J}{N}\gg\delta E,$$

where $\mathcal J$ is the strong-coupling scale and $\delta E$ is the exponentially fine level spacing of a particular finite-$N$ system in the relevant energy band. Schematically, $\delta E\sim e^{-S_*}/C$ when an entropy $S_*$ controls the number of levels in that band.

1. Infrared effective theory. The conformal and Schwarzian descriptions require $T,|\omega|\ll\mathcal J$ and a controlled large-$N$ expansion. This condition removes ordinary ultraviolet modes but does not decide whether the soft mode is classical or quantum.
2. Semiclassical Schwarzian saddle. When $CT\gg1$, equivalently $1\ll\beta\mathcal J\ll N$ for fixed $q$ and an order-one Schwarzian coefficient, fluctuations about the thermal saddle are small. This is the familiar window $\mathcal J/N\ll T\ll\mathcal J$.
3. Quantum Schwarzian. When $CT\lesssim1$, or roughly $\beta\mathcal J\gtrsim N$, one must integrate over $\operatorname{Diff}^{+}(S^1)/PSL(2,\mathbb R)$ rather than expand about a single $f(u)=u$ saddle. The Schwarzian theory has not failed; its quantum fluctuations have become leading.

At still lower energies, a fixed finite-$N$ Hamiltonian resolves individual levels and recurrence scales. Those effects are nonperturbative in the large-$N$ expansion and are not located by the scale $\mathcal J/N$. The exact Schwarzian partition function has the characteristic behavior

$$Z_{\mathrm{Sch}}(\beta) \propto \beta^{-3/2} \exp\!\left(\frac{2\pi^2C}{\beta}\right),$$

up to a $\beta$-independent normalization. Thus the $-\tfrac32\log\beta$ term is an order-$N^0$ quantum contribution, distinct from finite-coupling saddle corrections such as $O\!\left(N/(\beta\mathcal J)^2\right)$. The one-loop-exact Schwarzian result and density of states are derived in Stanford and Witten 2017, § 2.4, Eqs. (2.38)–(2.39).

There is also an important order of limits. The SYK residual entropy density $s_0$ is defined by taking $N\to\infty$ before $T\to0$. At fixed finite $N$, taking $T\to0$ instead probes the actual ground-state degeneracy, generally an $O(1)$ datum rather than $e^{Ns_0}$. This distinction prevents a large-$N$ entropy intercept from being misread as an exact finite-system degeneracy.

Thermodynamics fixes one coefficient, not the entropy constant

In the semiclassical overlap, write the leading partition function on either side as

$$\log Z_X(\beta) =-\beta E_{0,X}+S_{*,X} +\frac{2\pi^2C_X}{\beta} +\cdots, \qquad X\in\{\mathrm{JT},\mathrm{SYK}\}.$$

The energy zero $E_{0,X}$ must be retained until a subtraction convention is declared. On the JT side, $S_{*,\mathrm{JT}}=S_0$ is the coefficient of the topological disk weight and, in a top-down black-hole throat, may be inherited from an extremal entropy. On the SYK side, $S_{*,\mathrm{SYK}}=Ns_0+O(N^0)$ is the large-$N$ residual entropy in the order of limits just stated. Neither constant follows from the Schwarzian action. Maldacena and Stanford emphasize this separation immediately after their soft action Maldacena and Stanford 2016, text following Eq. (4.178), and § 5, Eqs. (5.180)–(5.183).

Now perform the thermodynamic derivatives rather than matching formulas by inspection:

$$\begin{aligned} F_X(T) &=-T\log Z_X =E_{0,X}-TS_{*,X}-2\pi^2C_XT^2+\cdots,\\ E_X-E_{0,X} &=-\frac{\partial}{\partial\beta}\log Z_X-E_{0,X} =2\pi^2C_XT^2+\cdots,\\ S_X-S_{*,X} &=\left(1-\beta\frac{\partial}{\partial\beta}\right)\log Z_X-S_{*,X} =4\pi^2C_XT+\cdots,\\ C_{V,X} &=\frac{\partial E_X}{\partial T} =4\pi^2C_XT+\cdots. \end{aligned}$$

Two useful checks appear immediately:

$$C_{V,X}=S_X-S_{*,X}+\cdots, \qquad E_X-E_{0,X}=\frac{1}{2}T\bigl(S_X-S_{*,X}\bigr)+\cdots.$$

Suppose an SYK calculation or simulation measures the intensive heat-capacity coefficient

$$\gamma_{\mathrm{SYK}} =\lim_{T/\mathcal J\to0} \lim_{N\to\infty} \frac{C_{V,\mathrm{SYK}}}{NT}$$

inside the semiclassical large-$N$ window. Equivalently, one may take a double scaling limit with $T/\mathcal J\to0$ while $NT/\mathcal J\to\infty$. The order of limits is part of the definition. Then

$$C_{\mathrm{SYK}} =\frac{N\gamma_{\mathrm{SYK}}}{4\pi^2}, \qquad \frac{\phi_r}{8\pi G_2} =\frac{N\gamma_{\mathrm{SYK}}}{4\pi^2}.$$

The entropy slope fixes $C$ and predicts the energy and heat-capacity coefficients without another fit. Matching $S_0$ to $Ns_0$, by contrast, is a separate input requiring microscopic or top-down information. It is not a Schwarzian derivation of black-hole microstates.

For disordered SYK, one must also say which thermodynamic object is evaluated. The usual large-$N$ saddle computes disorder-averaged quantities, often through a replica-diagonal approximation to the quenched free energy. Agreement between quenched and annealed answers at a displayed order does not make the operations identical, especially at very low temperature or in nonperturbative observables. The collective-field derivation and its averaging assumptions are developed in SYK Bilocal Collective Fields as Near-AdS₂ Data.

Bilocal probes require a matter dictionary

For an operator of infrared dimension $h$, the thermal bilocal dressed by $f$ has magnitude

$$\mathcal B_h(u_1,u_2;f) =\left[ \frac{f'(u_1)f'(u_2)} {\left[ (\beta/\pi)\sin\!\left( \pi\bigl(f(u_1)-f(u_2)\bigr)/\beta \right) \right]^2} \right]^h.$$

For a fermionic operator, the Euclidean-ordering sign must be supplied in addition to this magnitude. In conformal SYK the fundamental fermion has $\Delta_\chi=1/q$. In the convention above, its thermal saddle can be written

$$G_c(\tau) =b\,\operatorname{sgn}(\tau) \left[ \frac{\pi}{\beta\sin(\pi|\tau|/\beta)} \right]^{2\Delta_\chi}, \qquad \pi J^2 b^q =\left(\frac12-\Delta_\chi\right) \tan(\pi\Delta_\chi).$$

The normalization equation is an independent check that the same $J$ convention is being used. Under a reparametrization, the correlator becomes

$$G_f(u_1,u_2) =\bigl[f'(u_1)f'(u_2)\bigr]^{\Delta_\chi} G_c\bigl(f(u_1),f(u_2)\bigr).$$

Choosing a JT matter probe with $h=\Delta_\chi$ makes the two bilocals receive the same universal soft-mode dressing. This is a genuine test of the shared effective theory. It is not yet a complete operator dictionary: pure JT gravity contains no such matter field, and the match does not determine its normalization, interactions, higher-point couplings, or the full tower of SYK bilinears. Those data require a matter completion on the gravity side and a specified operator map. See JT Correlators, Bilocals, and Chaos for the gravity calculation and SYK Conformal Regime and Schwarzian Matching for the SYK calculation.

Chaos tests the same soft sector

A Lyapunov exponent is meaningful only after the correlator and contour are fixed. One useful regulated out-of-time-order correlator is

$$F(t)=\operatorname{Tr}\!\left[ yV(t)yW(0)yV(t)yW(0) \right], \qquad y=\frac{e^{-\beta H/4}}{Z^{1/4}}.$$

Subtract the disconnected contribution and choose operators that overlap with the reparametrization channel. In the leading strong-coupling, large-$N$ window, soft-mode exchange produces

$$F_{\mathrm{conn}}(t) \propto \frac{\beta}{C} \exp\!\left(\frac{2\pi t}{\beta}\right), \qquad t_{\mathrm{diss}}\ll t\ll t_*.$$

At fixed strong coupling, $t_*\sim(\beta/2\pi)\log(C/\beta)\sim(\beta/2\pi)\log N$ up to operator-dependent constants. The exponent $\lambda_L=2\pi/\beta$ saturates the chaos bound at leading order, but its equality on the two sides is another consequence of the same Schwarzian propagator. It is not independent evidence for a unique ultraviolet dual.

The gravity-side soft exchange and its second-sheet continuation are shown explicitly in Jensen 2016, Eqs. (32)–(38). That calculation also makes the matter-probe assumption visible: pure JT without probe operators does not define this OTOC.

Finite coupling gives a clean adversarial check. Take $N\to\infty$ first and then $q\to\infty$ while holding $\mathcal J$ fixed. In this controlled large-$q$ solution, define $v$ by

$$\beta\mathcal J =\frac{\pi v}{\cos(\pi v/2)}, \qquad 0<v<1,$$

as in Maldacena and Stanford 2016, § 2.4, Eqs. (2.16) and (2.19). The corresponding OTO kernel gives

$$\lambda_L=\frac{2\pi v}{\beta} <\frac{2\pi}{\beta}$$

at finite $\beta\mathcal J$. The maximal result is recovered only as $v\to1$ at strong coupling Maldacena and Stanford 2016, § 3.6, Eqs. (3.151)–(3.160). Operator prefactors, nonsoft channels, and the precise dissipation and scrambling windows also remain model-dependent.

Disorder averages, topology, and matrix ensembles are different objects

For one Hamiltonian realization,

$$Z_J(\beta)=\operatorname{Tr}e^{-\beta H_J}.$$

An average can be taken before or after the logarithm:

$$F_{\mathrm{quenched}} =-\frac{1}{\beta}\,\overline{\log Z_J}, \qquad F_{\mathrm{annealed}} =-\frac{1}{\beta}\log\overline{Z_J}.$$

These are different observables. Multi-copy moments such as $\overline{Z_J(\beta_1)Z_J(\beta_2)}$ contain still more information about sample-to-sample correlations.

On the gravity side, the disk Schwarzian path integral is a fixed-topology amplitude. Adding higher-genus JT surfaces defines a perturbative topology expansion. A matrix integral can reproduce that expansion order by order, but choosing such an ensemble and choosing its nonperturbative completion are additional steps. None of the equalities

$$Z_J \stackrel{?}{=} \overline{Z_J} \stackrel{?}{=} Z_{\mathrm{JT,\ disk}} \stackrel{?}{=} Z_{\mathrm{matrix\ ensemble}}$$

follows from $C_{\mathrm{JT}}=C_{\mathrm{SYK}}$. The perturbative JT/matrix-integral relation is established by Saad, Shenker, and Stanford 2019, §§ 3.4–3.5, especially Eqs. (6)–(7); its interpretation is separated carefully in JT Topological Expansion and Weil–Petersson Volumes and Non-Unique JT Matrix-Integral Completion.

Matched infrared data and their claim ceilings

Infrared quantities matched by the shared Schwarzian sector

Quantity	SYK definition	JT definition	Controlled comparison	Leading limitation
Soft action	Pseudo-Goldstone mode of the bilocal conformal saddle	Fluctuating regulated-boundary embedding	Same Schwarzian functional after quotienting by PSL(2,ℝ)	Finite-coupling and higher-derivative terms differ
Coefficient C	NαS(q)/J in the declared Majorana convention	φr/(8πG2) in the declared boundary convention	Equality fixes entropy slope, excess energy, and heat capacity	Does not fix either entropy intercept
Probe dressing	Reparametrized conformal bilocal of dimension h	Boundary bilocal for a chosen matter field of the same dimension	Same integral over the soft mode	Pure JT supplies no complete matter or operator dictionary
Leading chaos	Connected regulated OTOC in the soft channel	Crossed bilocal correlator with gravitational soft exchange	Same strong-coupling exponent and soft prefactor scaling	Finite-coupling exponent and operator prefactors are model-dependent

This table is deliberately narrower than an assertion of “SYK/JT duality.” The strongest common conclusion is that the displayed observables occupy the same controlled infrared universality sector. Each additional claim needs additional evidence.

Ultraviolet and finite-N adversaries

Two concrete comparisons show why the claim ceiling matters.

Change microscopic symmetry without changing the leading melonic limit. A disorder-free tensor Hamiltonian can reproduce the leading large-$N$ SYK correlation functions and thermodynamics while changing the microscopic symmetry, the presence of disorder, and the subleading expansion. This provides an explicit counterexample to the inference “same leading infrared saddle, therefore same microscopic theory” Witten 2019, §§ 1–3.

Keep the SYK ensemble but change $N$ modulo eight. For $q=4$, compare large-$N$ subsequences while holding the limiting intensive quantities $s_0$ and $C/N$ fixed. Within the appropriate fermion-parity blocks, $N\bmod8=0$ has GOE statistics, $N\bmod8=2$ or $6$ has GUE statistics, and $N\bmod8=4$ has GSE statistics. The leading large-$N$ saddle does not see this distinction, but the antiunitary symmetries, degeneracies, and finite-$N$ level correlations do. The pattern is derived in Cotler et al. 2017, § 3.1 and Appendix A. Leading thermodynamics therefore cannot determine the nonperturbative spectral symmetry class.

A second ultraviolet falsifier is even simpler: at $|\omega|\sim J$, the microscopic $-i\omega$ kinetic term can no longer be omitted from the Schwinger–Dyson equations, so the conformal propagator and the Schwarzian-only description fail. Infrared agreement makes no prediction for that crossover without further microscopic matching.

Data that the shared infrared sector does not determine

Data	Extra hypothesis that would be needed	Decisive test	Strongest claim after failure
Entropy constant	A microscopic count or top-down extremal-entropy map	Compare the large-N order of limits and the independently normalized disk weight	The temperature-dependent Schwarzian thermodynamics still matches
Microscopic Hilbert space	A state-by-state isomorphism and complete operator map	Compare dimensions, symmetry sectors, and exact matrix elements	A shared low-energy sector, not full-theory equivalence
Global symmetry and level class	Matching antiunitary symmetries, charges, and boundary conditions	Resolve degeneracies and unfolded finite-N level statistics	Leading large-N thermodynamics may remain universal
Complete operator spectrum	A specified JT matter completion and normalized source map	Compare nonsoft dimensions, OPE data, and higher-point functions	Only the chosen soft-dressed probes match
Averaging prescription	An equality between fixed-realization, disorder, topology, and matrix averages	Compare connected multi-copy moments and factorization	Single-boundary leading observables may still agree
Nonperturbative completion	A unique contour, spectrum, or matrix potential beyond the genus expansion	Compare exponentially small spectral data and exact factorization	The same perturbative Schwarzian or topological expansion

Evidence boundary and chapter-level maps

The comparison above uses the Schwarzian as a controlled low-energy effective theory. It does not assume that every SYK-like model has a simple local gravity dual, that every nearly AdS₂ throat has an SYK ultraviolet completion, or that a topology sum describes one fixed Hamiltonian. Modern reviews of the two sides, with complementary emphases, are Chowdhury et al. 2022, §§ V and XII.D–E and Mertens and Turiaci 2023, §§ 2–3.

The main remaining uncertainties are observable-dependent corrections beyond the leading Schwarzian, the choice of matter completion, the relation between disorder and gravitational averages, and the nonperturbative meaning of the gravitational path integral. Nearly AdS₂ Effective Theory Beyond the Leading Schwarzian organizes the first question; Random Matrices, Spectral Statistics, and Ensemble Questions and JT/SYK Spectral Form Factors and Universality Windows organize the late-time spectral questions.

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.

Common pitfalls

Calling the entire low-energy region “semiclassical.” The Schwarzian remains the appropriate effective theory when $CT\lesssim1$, but its path integral is then quantum rather than a small fluctuation about one saddle. Finite-$N$ discreteness occurs at a still finer scale.

Fitting $S_*$ and $C$ as though they were one datum. The constant entropy and the temperature-dependent Schwarzian term have different origins. Matching $C$ does not calculate $S_0$ or $Ns_0$.

Using maximal chaos as an independent confirmation. The leading exponent, entropy slope, and soft bilocal dressing all descend from the same reparametrization mode. They are valuable internal consistency checks, but not three independent proofs of a microscopic duality.

Writing “the SYK partition function” without an average. A fixed realization, an annealed average, a quenched average, and a multi-copy moment are different objects. State which one is meant before comparing it with a gravitational amplitude.

Treating pure JT as a complete bulk dual. Pure JT controls the gravitational soft sector. Correlators of nontrivial operators require matter fields, their boundary conditions, and a normalized source map.

Exercises

1. Derive the thermodynamic round trip

Starting from

$$\log Z=-\beta E_0+S_*+\frac{2\pi^2C}{\beta},$$

derive $F$, $E$, $S$, and $C_V$. Then eliminate $C$ to obtain two relations among $E-E_0$, $S-S_*$, and $C_V$.

Solution

Using $F=-\beta^{-1}\log Z$ gives

$$F=E_0-TS_*-2\pi^2CT^2.$$

Next,

$$E=-\partial_\beta\log Z =E_0+\frac{2\pi^2C}{\beta^2} =E_0+2\pi^2CT^2.$$

Either $S=-\partial F/\partial T$ or $S=(1-\beta\partial_\beta)\log Z$ gives

$$S=S_*+4\pi^2CT,$$

and differentiating the energy gives

$$C_V=\frac{\partial E}{\partial T}=4\pi^2CT.$$

Therefore

$$C_V=S-S_*, \qquad E-E_0=\frac{T}{2}(S-S_*)=\frac{T}{2}C_V$$

at this order. If any one of these relations fails in the proposed common window, one coefficient $C$ cannot describe all three observables.

2. Extract the gravity coupling from SYK heat capacity

Suppose a large-$N$ SYK calculation finds $C_V/N=\gamma T+O((T/J)^2)$. Derive the JT parameter combination fixed by the interface. State the conventions and limits needed for the answer to be meaningful.

Solution

Comparing with $C_V=4\pi^2C_{\mathrm{SYK}}T$ gives

$$C_{\mathrm{SYK}}=\frac{N\gamma}{4\pi^2}.$$

The interface then requires

$$\frac{\phi_r}{8\pi G_2} =\frac{N\gamma}{4\pi^2}, \qquad \frac{\phi_r}{G_2}=\frac{2N\gamma}{\pi}.$$

One must use the displayed Majorana anticommutator and coupling variance, the same definition of $J$, the JT convention $C_{\mathrm{JT}}=\phi_r/(8\pi G_2)$, and the large-$N$, strong-coupling semiclassical window $1\ll\beta\mathcal J\ll N$. The calculated heat capacity must also specify whether it is quenched, annealed, or fixed-realization. Dimensional analysis checks that $\gamma$, $C$, and $\phi_r/G_2$ all have inverse-energy units in these conventions.

3. Classify the infrared regimes

For fixed large $N$ and fixed $q$, classify the following thermal scalings: (a) $\beta\mathcal J=O(1)$, (b) $1\ll\beta\mathcal J\ll N$, (c) $\beta\mathcal J\sim N$, and (d) $\beta\delta E\gtrsim1$. Explain why $\beta\mathcal J\sim N$ is not automatically the discrete-spectrum regime.

Solution

(a) is the ultraviolet crossover, where the conformal approximation is not parametrically controlled. (b) is the semiclassical Schwarzian window: $T\ll\mathcal J$ but $CT\sim N/(\beta\mathcal J)\gg1$ up to a fixed $q$-dependent coefficient. (c) is the quantum-Schwarzian crossover, where $CT=O(1)$ and soft loops cannot be neglected. (d) resolves the individual levels of the finite-$N$ system.

The scales $C^{-1}\sim\mathcal J/N$ and $\delta E$ have different origins. The former marks quantum fluctuations of the collective soft mode; the latter is nonperturbatively small in the many-body entropy. There can therefore be a wide quantum-Schwarzian range between them.

4. Test a bilocal claim

A proposed dictionary matches the SYK fermion two-point function to a JT bilocal with $h=1/q$. List what this equality tests and what additional information would be required to promote it to a complete operator dictionary.

Solution

The equality tests the infrared scaling dimension and the universal transformation under the shared reparametrization mode. After integrating over $f$, it also tests the associated soft quantum corrections in a declared temperature range.

It does not fix the overall normalization, the fermionic ordering sign, the identity of a local bulk field, the bulk interaction vertices, the nonsoft bilinear tower, or higher-point OPE data. A complete dictionary would require a specified JT matter action, boundary conditions, a normalized source-to-operator map, and agreement of independent higher-point observables.

5. Apply the ultraviolet adversary

Suppose two models have the same leading $C/N$ and the same conformal fermion dimension, but one has quenched random couplings and the other is a disorder-free tensor model with a different global symmetry. Which claims survive, and which fail?

Solution

The shared leading entropy slope, heat capacity, soft bilocal dressing, and strong-coupling chaos channel can survive. These support membership in the same infrared universality sector for the tested observables.

Claims of identical microscopic Hamiltonians, identical global symmetry sectors, identical averaging prescriptions, identical subleading $1/N$ corrections, and a unique complete gravity dual fail. The strongest defensible conclusion is a partial low-energy correspondence, not an exact equivalence of full theories.

6. Keep four averages distinct

Define, in words, $Z_J$, $\overline{Z_J}$, $\overline{\log Z_J}$, and $\overline{Z_J(\beta_1)Z_J(\beta_2)}$. Which one is closest to a single JT disk amplitude, and what further assumption would still be required?

Solution

$Z_J$ is the partition function of one coupling realization. $\overline{Z_J}$ is its annealed disorder average. $\overline{\log Z_J}$ determines the quenched free energy. $\overline{Z_J(\beta_1)Z_J(\beta_2)}$ is a two-copy moment that can reveal correlations between spectra across the ensemble.

A single JT disk is most directly comparable to a one-boundary averaged partition function such as $\overline{Z_J}$ after normalizations and boundary conditions are matched. Even then, equality requires an explicit proposal identifying the gravitational path-integral measure with that disorder average; the Schwarzian coefficient alone does not supply it. A disk amplitude is not automatically a fixed-realization partition function or a quenched free energy.

References

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