Holography and Riemann 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.

1-loop partition function in AdS 3/CFT 2

“…At the asymptotic boundary, the configuration is a Riemann surface, which could be uniformized by the Schottky group. It was shown in [13] that the regularized semi-classical action of the handlebody solution could be described by a Liouville type action, the so-called Zograf-Takhtajan(ZT) action [14]. 1 Furthermore, the 1-loop partition function of the handlebody solutions has been conjectured to be [17] log Z| 1-loop = − γ ∞ m=2 log |1 − q m γ |, (1.2) where γ is the primitive conjugacy class of the Schottky group and q − 1 2 γ is the larger eigenvalue of γ.…”

“…In this section, we first give a brief review on Schottky uniformization, mainly basing on the work [13]. Then we discuss how to compute a higher genus partition function using the sewing prescription.…”

Higher spin entanglement and W N $$ {\mathcal{W}}_{\mathrm{N}} $$ conformal blocks

“…[26]). The bulk manifolds corresponding to these boundary topologies are handlebodies which can be described as a quotients of AdS 3 by Σ [27]. Roughly speaking, these gravitational saddles correspond to particular ways of "filling in" the boundary Riemann surface with Euclidean AdS 3 space.…”

Liouville action as path-integral complexity: from continuous tensor networks to AdS/CFT

“…If this derivation of time slice metric in AdS 3 really explains the mechanism of emergence of AdS in AdS/CFT, it should fit nicely with the dynamics of AdS gravity for the whole 3D space-time. One natural coordinate system in 3D gravity for our argument is as follows It is also useful to remember that connections between Liouville theory and 3D AdS gravity were discussed in earlier papers [59][60][61][62][63][64][65][66] (refer to [67] for a review). Especially the direct connection between the equation of motion in the SL(2, R) Chern-Simons gauge theory description of AdS gravity [68] and that of Liouville theory was found in [61] (see also closely related arguments [62][63][64][65]).…”