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Liouville action and Weil-Petersson metric on deformation spaces, global Kleinian reciprocity and holography

Leon A. Takhtajan, Lee-Peng Teo

TL;DR

The paper rigorously constructs the Liouville action S on deformation spaces of finitely generated, purely loxodromic Kleinian groups (including Fuchsian, quasi-Fuchsian, and Schottky cases) via homology–cohomology double complexes and shows P_F − P_QF = 1/2 ∂S, making −S a Kahler potential for the Weil–Petersson form. It proves a global Kleinian reciprocity, extends McMullen’s quasi-Fuchsian reciprocity, and establishes a holography principle: the regularized Einstein–Hilbert action in the bulk equals a modified Liouville action on the boundary, S[φ] − ∫ e^φ − 8π(2g−2) log 2. The work develops a robust deformation-theoretic framework, including Bers coordinates and WP geometry, and generalizes these results to Kleinian Groups of Class A, thereby unifying 2D Liouville theory, 3D gravity holography, and the deformation theory of Kleinian groups with a rigorous homological backbone.

Abstract

We rigorously define the Liouville action functional for finitely generated, purely loxodromic quasi-Fuchsian group using homology and cohomology double complexes naturally associated with the group action. We prove that the classical action - the critical point of the Liouville action functional, considered as a function on the quasi-Fuchsian deformation space, is an antiderivative of a 1-form given by the difference of Fuchsian and quasi-Fuchsian projective connections. This result can be considered as global quasi-Fuchsian reciprocity which implies McMullen's quasi-Fuchsian reciprocity. We prove that the classical action is a Kahler potential of the Weil-Petersson metric. We also prove that Liouville action functional satisfies holography principle, i.e., it is a regularized limit of the hyperbolic volume of a 3-manifold associated with a quasi-Fuchsian group. We generalize these results to a large class of Kleinian groups including finitely generated, purely loxodromic Schottky and quasi-Fuchsian groups and their free combinations.

Liouville action and Weil-Petersson metric on deformation spaces, global Kleinian reciprocity and holography

TL;DR

The paper rigorously constructs the Liouville action S on deformation spaces of finitely generated, purely loxodromic Kleinian groups (including Fuchsian, quasi-Fuchsian, and Schottky cases) via homology–cohomology double complexes and shows P_F − P_QF = 1/2 ∂S, making −S a Kahler potential for the Weil–Petersson form. It proves a global Kleinian reciprocity, extends McMullen’s quasi-Fuchsian reciprocity, and establishes a holography principle: the regularized Einstein–Hilbert action in the bulk equals a modified Liouville action on the boundary, S[φ] − ∫ e^φ − 8π(2g−2) log 2. The work develops a robust deformation-theoretic framework, including Bers coordinates and WP geometry, and generalizes these results to Kleinian Groups of Class A, thereby unifying 2D Liouville theory, 3D gravity holography, and the deformation theory of Kleinian groups with a rigorous homological backbone.

Abstract

We rigorously define the Liouville action functional for finitely generated, purely loxodromic quasi-Fuchsian group using homology and cohomology double complexes naturally associated with the group action. We prove that the classical action - the critical point of the Liouville action functional, considered as a function on the quasi-Fuchsian deformation space, is an antiderivative of a 1-form given by the difference of Fuchsian and quasi-Fuchsian projective connections. This result can be considered as global quasi-Fuchsian reciprocity which implies McMullen's quasi-Fuchsian reciprocity. We prove that the classical action is a Kahler potential of the Weil-Petersson metric. We also prove that Liouville action functional satisfies holography principle, i.e., it is a regularized limit of the hyperbolic volume of a 3-manifold associated with a quasi-Fuchsian group. We generalize these results to a large class of Kleinian groups including finitely generated, purely loxodromic Schottky and quasi-Fuchsian groups and their free combinations.

Paper Structure

This paper contains 35 sections, 27 theorems, 307 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Liouville action functional
  3. Homology and cohomology set-up
  4. The Fuchsian case
  5. Homology computation
  6. Cohomology computation
  7. The action functional
  8. The quasi-Fuchsian case
  9. Homology construction
  10. Cohomology construction
  11. The Liouville action functional
  12. Deformation theory
  13. The deformation space
  14. Variational formulas
  15. Variation of the classical action
  16. ...and 20 more sections

Key Result

Lemma 2.4

For all $\phi\in\mathcal{CM}(X)$ and $\sigma\in C^\infty(X,\mathbb{R})$, where $d^2z=dx\wedge dy$ is the Lebesgue measure and $K=-2e^{-\phi}\phi_{z\bar{z}}$ is the Gaussian curvature of the metric $e^{\phi}|dz|^2$.

Figures (1)

  • Figure 1: Conventions for the fundamental domain $F$

Theorems & Definitions (79)

  • Remark 2.1
  • Remark 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Corollary 2.5
  • proof
  • Remark 2.6
  • Lemma 2.7
  • proof
  • ...and 69 more