JT Gravity and the Schwarzian Boundary Mode

Jackiw–Teitelboim (JT) gravity fixes the bulk metric to constant negative curvature while allowing a dilaton to measure departures from an extremal throat. After the variationally complete action is evaluated on a regulated boundary curve, the surviving degree of freedom is a time reparametrization with Schwarzian action. Its coupling and validity range come from the parent black hole; JT is not by itself a microscopic completion.

Required background. AdS2 Boundary Conditions and Fragmentation supplies the constrained boundary data. Effective Field Theory as a Controlled Expansion supplies the low-energy logic.

Helpful background. Field Variations and Boundary Terms supplies the variational calculation. Boundaries, Surface Counterterms, and Boundary Stress supplies renormalized boundary observables.

Evidence cutoff: 25 July 2026.

The JT variational problem

In Euclidean signature and units $L_2=1$, take

$$I_{\mathrm{JT}} =-\frac{\Phi_0}{16\pi G_2} \left[\int_M\sqrt g\,R+2\int_{\partial M}K\right] -\frac{1}{16\pi G_2}\int_M\sqrt g\,\varphi(R+2) -\frac{1}{8\pi G_2}\int_{\partial M}\sqrt h\,\varphi(K-1).$$

The first line is topological and contributes $-S_0\chi$ with $S_0=\Phi_0/(4G_2)$. The subtraction in the last term makes the asymptotic variational problem finite. Varying $\varphi$ gives

$$R=-2,$$

while varying the metric determines the dilaton on that constant-curvature geometry. There is no propagating bulk graviton.

From the cutoff curve to the Schwarzian

Write Euclidean AdS₂ as $ds^2=(dt^2+dz^2)/z^2$. Impose

$$\left.ds^2\right|_{\partial M}=\frac{du^2}{\epsilon^2}, \qquad \varphi\big|_{\partial M}=\frac{\varphi_r}{\epsilon}.$$

Parameterizing the curve by $t=f(u)$ gives $z=\epsilon f'+O(\epsilon^3)$. Its extrinsic curvature is

$$K=1+\epsilon^2\{f,u\}+O(\epsilon^4), \qquad \{f,u\}=\frac{f'''}{f'}-\frac32\left(\frac{f''}{f'}\right)^2.$$

The renormalized boundary action becomes

$$I_{\mathrm{Sch}}[f] =-C\int_0^\beta du\, \left\{\tan\frac{\pi f(u)}{\beta},u\right\}, \qquad C=\frac{\varphi_r}{8\pi G_2}.$$

The quotient by $SL(2,\mathbb R)$ removes reparametrizations that leave the AdS₂ geometry unchanged. This derivation and its relation to near-extremal black holes are given by Maldacena, Stanford, and Yang 2016.

Thermal saddle as the first application

For $f(u)=u$,

$$\left\{\tan\frac{\pi u}{\beta},u\right\} =\frac{2\pi^2}{\beta^2}, \qquad \log Z(\beta)=S_0+\frac{2\pi^2C}{\beta}+\cdots .$$

Therefore

$$E-E_0=2\pi^2CT^2, \qquad S=S_0+4\pi^2CT, \qquad C_V=4\pi^2CT.$$

All three coefficients are linked. Matching only the entropy slope while missing the energy coefficient signals a normalization or ensemble error. The exact Schwarzian path integral refines the ellipsis with a one-loop power of $\beta$ Stanford and Witten 2017.

Breakdown above the throat scale

Increase $T$ until the proper excitation probes the end of the AdS₂ throat. Kaluza–Klein fields, matter modes, nonlinearities of the dilaton potential, and the asymptotic region then contribute at the same order as the Schwarzian. A good one-observable fit cannot suppress them.

The controlled statement is $T,\omega\ll\Lambda_{\mathrm{throat}}$ with $C$ matched to a specified parent solution and with $1/C$ corrections tracked. Finite $C$ Schwarzian quantum mechanics still does not choose a unique ultraviolet Hamiltonian or a unique nonperturbative gravitational completion.

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

• Maldacena, Juan, Douglas Stanford, and Zhenbin Yang. “Conformal Symmetry and Its Breaking in Two-Dimensional Nearly Anti-de Sitter Space.” Progress of Theoretical and Experimental Physics 2016, 12C104 (2016). DOI.
• Stanford, Douglas, and Edward Witten. “Fermionic Localization of the Schwarzian Theory.” Journal of High Energy Physics 2017, 008 (2017). DOI.