Epstein curves and holography of the Schwarzian action

Abstract

Inspired by the duality between Jackiw-Teitelboim gravity and Schwarzian field theory, we show identities between the Schwarzian action of a circle diffeomorphism and (1) the length and the hyperbolic area enclosed by the Epstein curve in the hyperbolic disk $\mathbb{D}$, (2) the asymptotic excess in the isoperimetric inequality for the equidistant Epstein foliation, (3) the variation of the Loewner energy along its equipotential foliation, and (4) the asymptotic change in hyperbolic area under a conformal distortion near the circle. Together with the isoperimetric inequality and the monotonicity of the Loewner energy, our identities immediately imply two new proofs of the non-negativity of the Schwarzian action for circle diffeomorphisms. Moreover, we extend the identity (1) to the dual Epstein curve in de Sitter $2$-space, to the Schwarzian action for the coadjoint orbits $\mathrm{Möb}_n (\mathbb{S}^1) \backslash \mathrm{Diff}(\mathbb{S}^1)$, and to piecewise Möbius circle diffeomorphisms which are only $C^{1,1}$ regular. We also show that the horocycle truncation used in the construction of the Epstein curve defines a renormalized length of hyperbolic geodesics in $\mathbb{D}$, which coincides with the log of the bi-local observable. From this, we show that the bi-local observables on the edges of any ideal triangulation of $\mathbb{D}$ determine the circle diffeomorphism.

Figures (7)

• Figure 1: Illustration of part of the Epstein curves associated with a metric $h$ (orange) and $2h$ (blue), where $H_z$ is a horocycle determined by $h$ at $z$, the Epstein curve is the envelope of the family of corresponding horocycles $(H_z)_{z \in \mathbb{S}^1}$, and $\widetilde{\operatorname{Ep}}_h (z) \in T^1 \mathbb{D}$ is the outward unit normal with base point $\operatorname{Ep}_h (z) \in \mathbb{D}$. The hyperbolic length of the red geodesic segment gives a positive renormalized length between $z, w \in \mathbb{S}^1$ as in Proposition \ref{['prop:intro_length']}.
• Figure 2: Left: Epstein curve (orange) and the horocycles (blue) associated with the metric $h = \varphi_* \mathrm{d} \theta$, where $\varphi(\theta) = \frac{1}{2}\sin(\theta) + \theta$. Right: Epstein curve and the horocycles associated with the metric $e^{1/2} h$. Middle: the family of Epstein curves associated with the metrics $\{h_t = \mathrm{e}^{t} h\}_{t \ge 0}$ which form a foliation near $\mathbb{S}^1$.
• Figure 3: The horocycles and Epstein curve for $\varphi_*\mathrm{d} \theta$, where $\varphi= (\phi(z))^n$, and $\phi^{-1}(\theta) = \frac{1}{2} \sin(\theta) + \theta$, for $n$ from $1$ to $8$.
• Figure 4: On the right are the Epstein curve and horocycles for the double cover corresponding to the metric $h^a$, $a=1/2$ in Example \ref{['ex:z3_conjugated']} (i.e., $h^{1/2}$ pushed forward by $z^2$). On the left are twenty five evenly-space horocycles for only $\theta\in [0,\pi/2]$, with the point on the Epstein curve marked on them.
• Figure 5: Example of an ideal triangulation, where only finitely many edges are drawn.

Theorems & Definitions (100)

• Theorem 1.1: See Theorem \ref{['thm:SchArea']}
• Lemma 1.2: See Lemma \ref{['lem:asymptoticLA']}
• Corollary 1.3
• Theorem 1.4: See Theorem \ref{['thm:excess']} and Corollary \ref{['cor:non-negativeSch_isoperimetric']}
• Proposition 1.5: See Proposition \ref{['prop:bilocal_is_length']}
• Theorem 1.6: See Theorem \ref{['thm:IschnLength']}
• Theorem 1.7: See Theorem \ref{['thm:var_IL_Sch']}
• Remark 1.8: Second proof of non-negativity of $I_{\operatorname{Sch}}$
• Theorem 1.9: See Theorem \ref{['thm:conformal_distortion']}
• Corollary 1.10: See Corollary \ref{['cor:conformal_distortion_area']}