The Schwarzian Boundary Mode

The low-energy mode of a nearly AdS2 throat is a boundary reparametrization, but three of its apparent directions are gauge-equivalent. Evaluating the JT boundary term on a regulated curve produces the Schwarzian action on $\mathrm{Diff}(S^1)/\mathrm{SL}(2,\mathbb R)$. This page derives that quotient, evaluates the thermal saddle, and connects its coefficient to the near-extremal entropy rather than treating the Schwarzian as an arbitrary one-dimensional model.

Required background. Gauge Theory and String-Theory Preparation provides the effective-action and symmetry-normalization language used below.

Helpful background. Quantum Gravity supplies the effective-boundary description; Quantum Information supplies thermal and spectral language; Semiclassical Gravity provides Euclidean thermodynamics; and Black-Hole Information motivates the late-time limitations.

The regulated boundary curve

Use Euclidean Poincaré AdS2,

$$ds^2=L_2^2\frac{dt^2+dz^2}{z^2},$$

and parametrize the cutoff curve by $(t,z)=(f(u),z(u))$. Fix its induced metric and dilaton by

$$\frac{L_2^2}{z^2}\left(f'^2+z'^2\right)=\frac{L_2^2}{\epsilon^2}, \qquad \phi_b=\frac{\phi_r}{\epsilon}.$$

For an orientation-preserving curve with $f'\gt0$, solving the first condition gives $z=\epsilon f'+O(\epsilon^3)$. With $K=h^{\mu\nu}\nabla_\mu n_\nu$ and the normal pointing toward the asymptotic boundary at smaller $z$, the extrinsic curvature expands as

$$K=\frac{1}{L_2}\left[1+\epsilon^2\{f,u\}+O(\epsilon^4)\right], \qquad \{f,u\}=\frac{f'''}{f'}-\frac{3}{2}\left(\frac{f''}{f'}\right)^2 .$$

The counterterm cancels the leading $1/\epsilon^2$ contribution. The finite part of the same JT boundary term is therefore

$$I_{\rm Sch}[f]=-C\int du\,\{f,u\}, \qquad C=\frac{\phi_r}{8\pi G_2}.$$

This derivation, including the cutoff curve and thermal evaluation, appears in Maldacena, Stanford, and Yang 2016, §3, Eqs. (3.15)–(3.21), pp. 8–10. The sign assumes the Euclidean convention $Z\simeq e^{-I_E}$ and positive $C$ inherited from a sensible near-extremal heat capacity.

Why the quotient is by SL(2,R)

The Schwarzian is invariant under a projective transformation

$$f(u)\longmapsto\frac{a f(u)+b}{c f(u)+d}, \qquad ad-bc=1.$$

These three parameters are the exact projective isometries of AdS2, so they do not label distinct boundary configurations. On the thermal circle the actual monotonic circle map is $\varphi(u+\beta)=\varphi(u)+\beta$, with $\varphi'\gt0$; its projective coordinate is $f(u)=\tan[\pi\varphi(u)/\beta]$. The poles of this coordinate do not mean that the circle map is singular. The physical integration space is conventionally written $\mathrm{Diff}(S^1)/\mathrm{SL}(2,\mathbb R)$ (the central $\pm I$ act identically). Infinitesimally, the thermal saddle has three zero modes corresponding to Fourier numbers $0$ and $\pm1$.

This quotient matters quantitatively. Counting the zero modes incorrectly changes the one-loop measure and the power of $\beta$ in the partition function. Stanford and Witten 2017, §§2–3, pp. 5–16 formulate the symplectic measure and show why the Schwarzian path integral is one-loop exact in its stated setting. That result concerns this effective theory; it is not a proof that an ultraviolet black-hole theory has only the Schwarzian sector.

The quotient is also a useful diagnostic of conventions. A calculation performed on the line may fix three points, whereas a thermal-circle calculation removes three Fourier modes; these are two gauge choices for the same projective redundancy. Dividing only by the compact time-translation subgroup leaves two spurious zero modes. Dividing by a larger group would erase genuine boundary fluctuations and would spoil the thermodynamic density of states.

Thermal saddle and thermodynamic check

On a thermal circle of length $\beta$, a convenient projective representative is

$$f_\beta(u)=\tan\frac{\pi u}{\beta}, \qquad \{f_\beta,u\}=\frac{2\pi^2}{\beta^2}.$$

The classical contribution is $\log Z=S_0+2\pi^2C/\beta$. Differentiation gives

$$E=-\partial_\beta\log Z=\frac{2\pi^2C}{\beta^2}, \qquad S=(1-\beta\partial_\beta)\log Z =S_0+\frac{4\pi^2C}{\beta}.$$

Substituting $C=\phi_r/(8\pi G_2)$ reproduces the JT result $S-S_0=\pi\phi_rT/(2G_2)$. This is an independent normalization check tying together the extrinsic-curvature expansion, thermal period, and horizon entropy. The quadratic fluctuation operator has positive nonzero eigenvalues for $C\gt0$ after the three projective directions are removed.

Correlators and limits of the soft sector

A matter correlator is dressed by replacing its conformal time separation with $f(u)$ and integrating over the soft mode. This captures leading gravitational time reparametrization and, in a controlled window, the enhanced four-point function associated with chaos. It does not include extra light bulk fields, Kaluza–Klein modes, finite-throat matching, or arbitrary high-frequency probes. The effective description requires energies below the throat gap and fluctuations small enough that omitted irrelevant operators remain suppressed.

At very late times, the Schwarzian spectral density is an effective, smoothed object. Whether a particular fixed microscopic theory has a discrete spectrum, an ensemble interpretation, or additional sectors is not fixed by the boundary action alone. Those distinctions become central on the JT spectral-statistics page.

The boundary conditions and approximation regime must therefore be stated together: fixed proper boundary time, fixed renormalized dilaton value, positive $C$, energies below the first omitted throat mode, and leading order in the irrelevant couplings that connect the throat to its exterior. The small parameter is not simply $G_2$. A semiclassical saddle can coexist with a failure of the one-field truncation if another mode becomes light. Conversely, a valid Schwarzian effective theory can include its own quantum fluctuations without claiming that all ultraviolet gravitational loops are negligible.

One final check is reparametrization covariance of observables. A bilocal of dimension $\Delta$ transforms with factors $[f'(u_1)f'(u_2)]^\Delta$ as well as the transformed separation. Retaining the separation but dropping the Jacobians gives a result that is neither projectively invariant nor correctly normalized. This check distinguishes genuine soft-mode dressing from an arbitrary change of the time coordinate.

Exercises

Show that the thermal representative has a constant Schwarzian and identify the three infinitesimal directions that must be removed from the fluctuation determinant.

Solution

Direct differentiation of $f_\beta=\tan(\pi u/\beta)$ gives $\{f_\beta,u\}=2\pi^2/\beta^2$, so its action is proportional to $\beta$ times this constant and hence to $1/\beta$. A projective change generated by $1$, $f$, or $f^2$ leaves the Schwarzian unchanged. Around the thermal circle these become the Fourier modes $n=0,\pm1$. They are exact AdS2 isometries, not physical Gaussian fluctuations, so including them would make the determinant singular and would overcount three gauge-equivalent configurations.

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 (2016) 12C104. arXiv:1606.01857.
• Stanford, Douglas, and Edward Witten. “Fermionic Localization of the Schwarzian Theory.” Journal of High Energy Physics 10 (2017) 008. arXiv:1703.04612.