[Paper Review] Holography and Riemann Surfaces

Kirill Krasnov|ArXiv.org|May 11, 2000

Black Holes and Theoretical Physics19 references5 citations

TL;DR

This paper extends holography in 2+1 dimensions to asymptotically AdS spaces with arbitrary genus Riemann surface boundaries by constructing such spaces via discrete identifications of Euclidean AdS₃ using classical Schottky groups. It shows that the regularized gravitational action precisely reproduces the Liouville action on Riemann surfaces as defined by Takhtajan and Zograf, establishing a bulk/boundary correspondence that links Teichmüller theory to holographic thermodynamics and provides a classical realization of boundary CFT partition functions on higher-genus surfaces.

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.

Motivation & Objective

• To generalize holography in 2+1 dimensions beyond simple boundary topologies to include arbitrary genus Riemann surfaces as conformal boundaries.
• To construct Euclidean AdS₃ spaces with non-trivial topology via discrete identifications using classical Schottky groups.
• To demonstrate that the regularized on-shell gravitational action for these spaces matches the Liouville action on Riemann surfaces as defined by Takhtajan and Zograf.
• To interpret the resulting geometry in terms of thermodynamics and derive the boundary CFT partition function in the semi-classical approximation.
• To connect the mathematical structures of Teichmüller and Schottky spaces to physical holographic observables such as thermodynamic parameters and correlation functions.

Proposed method

• Constructing asymptotically AdS spaces with arbitrary genus Riemann surface boundaries by quotienting Euclidean AdS₃ using classical Schottky groups.
• Using the semi-classical approximation to the gravity path integral, where the partition function is determined by the on-shell action of classical solutions matching a given boundary metric.
• Regularizing the gravitational action via volume renormalization to obtain a finite, well-defined quantity for each such space.
• Identifying the regularized action with the Liouville action on Riemann surfaces, as defined by Takhtajan and Zograf, using the pairing between Beltrami differentials and quadratic differentials.
• Leveraging the Weyl-Petersson metric and Teichmüller theory to interpret the moduli space of Riemann surfaces as the parameter space for these bulk geometries.
• Mapping the Schottky group parameters to thermodynamic variables, where the coefficients $ c_i^ u $ in the Schwarzian derivative expansion play the role of extensive parameters.

Experimental results

Research questions

• RQ1 Can holography be meaningfully extended to asymptotically AdS spaces whose conformal boundary is a Riemann surface of arbitrary genus?
• RQ2 How does the regularized gravitational action for such spaces relate to known actions in two-dimensional conformal field theory?
• RQ3 What is the physical interpretation of the Teichmüller and Schottky spaces in the context of holographic gravity and thermodynamics?
• RQ4 How do the parameters of the Schottky group correspond to thermodynamic quantities in the boundary CFT?
• RQ5 Can the boundary CFT partition function be derived from the bulk gravity path integral in this generalized holographic setting?

Key findings

• The regularized on-shell gravitational action for Euclidean AdS₃ spaces with arbitrary genus Riemann surface boundaries exactly reproduces the Liouville action on Riemann surfaces as defined by Takhtajan and Zograf.
• The construction realizes these spaces as 'analytic continuations' of Lorentzian black hole solutions with internal wormhole topologies, linking Euclidean and Lorentzian holography.
• The parameters $ c_i^ u $ in the expansion of the Schwarzian derivative of the uniformizing map correspond to extensive thermodynamic parameters, while Schottky coordinates $ ho_i $ serve as intensive parameters.
• The boundary CFT partition function is fully determined by the bulk gravity path integral in the semi-classical limit, with the result being equivalent to the partition function of Liouville theory on Riemann surfaces.
• The Weyl-Petersson metric on Teichmüller space emerges naturally from the geometry of the bulk, providing a Kähler structure that is invariant under the modular group.
• The map from Teichmüller space to Schottky space allows the lifting of vector fields and quadratic differentials, enabling a geometric realization of the dual CFT's moduli dependence.