Holography and Riemann surfaces

By K. Krasnov. 2000

231 citations recorded by the source corpus.

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.

Subfields: Astronomy and Astrophysics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/38a7eab901863fef4223c427e1408bce87868da8

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Astronomy and Astrophysics

1. V. Hubeny
2. D. Harlow
3. R. Myers
4. M. Rangamani
5. Thomas Hartman
6. J. Maldacena
7. D. Stanford
8. T. Takayanagi

In Nuclear and High Energy Physics

1. V. Hubeny
2. D. Harlow
3. R. Myers
4. Thomas Hartman
5. J. Maldacena
6. M. Rangamani
7. T. Takayanagi
8. D. Stanford

In Statistical and Nonlinear Physics

1. D. Harlow
2. R. Myers
3. V. Hubeny
4. Thomas Hartman
5. D. Stanford
6. J. Maldacena
7. M. Rangamani
8. S. Shenker

Citations in the corpus

• The renormalized volume and the volume of the convex core of quasifuchsian manifolds (2011, result): “This renormalized volumes turns out to be strongly related to the Liouville functional previously studied by Takhtajan and Zograf [21], and Takhtajan and Teo [22], see [13].”
• Inflation in AdS/CFT (2005, background): “A second example of a spacetime with causally disconnected regions that appears to be described by a pure state involves the “Swedish Geons” of [68,69,70,28,71,72].”
• Entanglement Entropy at Large Central Charge (2013, background): “The on-shell Einstein action on a 3-manifold M3 constructed in this way is equal to an appropriately defined Liouville action on M2 [20, 21, 22, 12],”
• Entanglement Renyi entropies in holographic theories (2010, background): “For holographic theories, despite considerable progress (especially in the context of pure gravity theories), explicit formulas are not available, even in the largec limit; see for example [30, 31, 32, 33, 34].”
• Holographic entanglement beyond classical gravity (2013, background): “[18, 5], the bulk action of PSL(2,C) maps hemispheres to hemispheres.”
• Geometric aspects of the AdS/CFT correspondence (2004, unknown): “itational action and its stressenergy tensor on Esd or Easd appears to closely resemble the identification of the renormalized action with the Liouville action on the boundary in dimensions 2 + 1, cf. [35], [19]. It would be very interesting to pursue this analogy further. The discussion above sugges”
• One-loop Partition Functions of 3D Gravity (2008, background): “These handlebodies may be thought of as the higher genus generatizations of the BTZ black hole [16, 18].”
• AdS2 holographic dictionary (2016, background): “18), both u+ and uo need to be expressed in terms of a suitable ‘uniformizing’ Liouville field as is done in the case of pure AdS3 gravity in [68].”
• Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial (2003, background): “These manifolds also have a nice interpretation in three-dimensional gravity [ 30 ], and will”
• Wormholes in AdS (2004, background): “This is a geometry that results by quotienting H3 by a so called Schottky group, see [33] for further discussion (10).”
• Minimal surfaces and particles in 3-manifolds (2005, unknown): “sults give an interesting perspective on the question of “analytic continuation” between different constant curvature 3-manifolds. Such an analytic continuation was considered by one of the authors in [Kra00] as a continuation between the negative curvature Lorentzian and Euclidean signature cases, f”
• The Entanglement Renyi Entropies of Disjoint Intervals in AdS/CFT (2013, methodology): “Following [27] we need to pick a regularization surface in order to define the bulk action.”