Holography and Riemann Surfaces

TL;DR

This work extends holography to asymptotically $AdS_3$ spacetimes with arbitrary boundary topology by quotienting $AdS_3$ with classical Schottky groups, relating bulk geometry to boundary Riemann surfaces. The semiclassical gravity action is computed as a regularized volume, which exactly reproduces the Takhtajan–Zograf Liouville action on the boundary, with central charge $c=\frac{3l}{2G}$; this underpins a boundary CFT description. The boundary theory is explored via a modular sum over Schottky space, linking the bulk geometry to Liouville theory and the Schwarzian stress-energy of the uniformization map, yielding a thermodynamic picture on the Schottky/Teichmüller moduli. The framework suggests a path to quantizing 2+1 gravity through Liouville theory and Teichmüller/Schottky space geometry, with phase transitions between Euclidean AdS and black-hole geometries at higher genus reflecting rich boundary CFT structure.

Abstract

We study holography for asymptotically AdS spaces with an arbitrary genus compact Riemann surface as the conformal boundary. Such spaces can be constructed from the Euclidean AdS_3 by discrete identifications; the discrete groups one uses are the so-called classical Schottky groups. As we show, the spaces so constructed have an appealing interpretation of ``analytic continuations'' of the known Lorentzian signature black hole solutions; it is one of the motivations for our generalization of the holography to this case. We use the semi-classical approximation to the gravity path integral, and calculate the gravitational action for each space, which is given by the (appropriately regularized) volume of the space. As we show, the regularized volume reproduces exactly the action of Liouville theory, as defined on arbitrary Riemann surfaces by Takhtajan and Zograf. Using the results as to the properties of this action, we discuss thermodynamics of the spaces and analyze the boundary CFT partition function. Some aspects of our construction, such as the thermodynamical interpretation of the Teichmuller (Schottky) spaces, may be of interest for mathematicians working on Teichmuller theory.

Figures (10)

• Figure 1: Genus one case.
• Figure 2: Genus two case.
• Figure 3: BTZ black hole: the initial slice geometry.
• Figure 4: BTZ black hole: the spacetime picture. Two asymptotic regions are shown: they are parts of the boundary cylinder lying between the timelike geodesics. The future event horizon, which is the boundary of the past of the asymptotic infinity, is shown. Note that it intersects the initial slice along the minimal line connecting the two geodesics bounding the fundamental region.
• Figure 5: Initial slice geometry of the single asymptotic region black hole with a toroidal wormhole inside the horizon