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**Holography and Riemann surfaces.**
*(English)*
Zbl 1011.81068

Summary: 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 \(\text{AdS}_3\) 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 Teichmüller (Schottky) spaces, may be of interest for mathematicians working on Teichmüller theory.

### MSC:

81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |

83C57 | Black holes |

30F10 | Compact Riemann surfaces and uniformization |

81S40 | Path integrals in quantum mechanics |

32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |

14J80 | Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants) |

83-02 | Research exposition (monographs, survey articles) pertaining to relativity and gravitational theory |

83C47 | Methods of quantum field theory in general relativity and gravitational theory |