Epstein curves and holography of the Schwarzian action

Abstract

We apply Epstein's construction of hypersurfaces in the hyperbolic disk $\mathbb D$ to prove identities between the Schwarzian action on $\operatorname{PSL}_2(\mathbb R)\backslash \mathrm{Diff}^3 (\mathbb S^1)$, the length of the corresponding Epstein curve in $\mathbb D$, and the area enclosed by the Epstein curve. These results are inspired by the holographic duality between Jackiw--Teitelboim gravity and Schwarzian field theory. 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 logarithm of the bi-local observable of Schwarzian field theory. The construction of the Epstein curve also extends to the coadjoint orbits $\operatorname{PSL}_2^{(n)}(\mathbb R)\backslash \mathrm{Diff}^3 (\mathbb S^1)$, and we obtain the same identities for the analog of the Schwarzian action on these coadjoint orbits. Furthermore, we show that the Schwarzian action is the derivative of the Loewner energy of the welded Jordan curve. This energy is the action functional of Schramm--Loewner evolutions and holographically expressed as a renormalized volume in hyperbolic $3$-space. As a by-product of these relations, we obtain two immediate proofs of the non-negativity of the Schwarzian action using the isoperimetric inequality and the monotonicity of the Loewner energy.