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

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.

60 pages, proof of the Lemma 5.1 corrected, references and section 5.3 added

Cited by in corpus (25)

• Semiclassical 3D gravity as an average of large-c CFTs
• On the renormalized volume of hyperbolic 3-manifolds
• Minimal surfaces and particles in 3-manifolds
• Summing over Geometries in String Theory
• Partition functions of the tensionless string
• No Ensemble Averaging Below the Black Hole Threshold
• Holographic Calculation for Large Interval Rényi Entropy at High Temperature
• Quantum Liouville theory in the background field formalism I. Compact Riemann surfaces
• Uniformization, Calogero-Moser/Heun duality and Sutherland/bubbling pants
• One loop partition function in AdS_3/CFT_2
• The Liouville Geometry of N=2 Instantons and the Moduli of Punctured Spheres
• Schwarzian derivatives, projective structures, and the Weil-Petersson gradient flow for renormalized volume
• Closed String Partition Functions in Toroidal Compactifications of Doubled Geometries
• The complex symplectic geometry of the deformation space of complex projective structures
• The Weil-Petersson gradient flow of renormalized volume and 3-dimensional convex cores
• Liouville action and Holography on quasi-Fuchsian deformation spaces
• A symplectic map between hyperbolic and complex Teichmüller theory
• Holomorphic factorization of determinants of Laplacians using quasi-Fuchsian uniformization
• The convex core of quasifuchsian manifolds with particles
• Classical Liouville Action and Uniformization of Orbifold Riemann Surfaces
• On hermitian-holomorphic classes related to uniformization, the dilogarithm, and the Liouville Action
• One-Loop Partition Function, Gauge Accessibility and Spectra in AdS${}_3$ Gravity
• Cylinder transition amplitudes in pure AdS${}_3$ Gravity
• A different expression of the Weil-Petersson potential on the quasi-Fuchsian deformation space
• Liouville Action for Harmonic Diffeomorphisms