Liouville action and Holography on quasi-Fuchsian deformation spaces

TL;DR

This paper explores the Liouville action in the context of quasi-Fuchsian groups, establishing formulas for its variations, relating it to holography principles, and constructing potential functions for associated Kähler metrics.

Contribution

It provides new formulas for the Liouville action involving elliptic elements and establishes a holography relation with renormalized volume in quasi-Fuchsian spaces.

Findings

Derived the elliptic contribution to Liouville action in terms of Bloch-Wigner functions.

Proved variation formulas for the Liouville action on deformation spaces.

Established a holography principle linking Liouville action and renormalized volume.

Abstract

We study the Liouville action for quasi-Fuchsian groups with parabolic and elliptic elements. In particular, when the group is Fuchsian, the contribution of elliptic elements to the classical Liouville action is derived in terms of the Bloch-Wigner functions. We prove the first and second variation formulas for the classical Liouville action on the quasi-Fuchsian deformation space. We prove an equality expressing the holography principle, which relates the Liouville action and the renormalized volume for quasi-Fuchsian groups with parabolic and elliptic elements. We also construct the potential functions of the K\"ahler forms corresponding to the Takhtajan-Zograf metrics associated to the elliptic elements in the quasi-Fuchsian groups.