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Liouville Action and Holography on Quasi-Fuchsian Deformation Spaces

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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 Kähler forms corresponding to the Takhtajan–Zograf metrics associated to the elliptic elements in the quasi-Fuchsian groups.

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Authors and Affiliations

  1. School of Mathematics, Korea Institute for Advanced Study, 207-43, Hoegiro 85, Dong-daemun-gu, Seoul, 130-722, Korea

    Jinsung Park

  2. Department of Mathematics, Xiamen University Malaysia, Jalan Sunsuria, 43900, Bandar Sunsuria, Selangor, Malaysia

    Lee-Peng Teo

Authors
  1. Jinsung Park
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  2. Lee-Peng Teo
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Correspondence to Jinsung Park.

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Communicated by X. Yin

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Park, J., Teo, LP. Liouville Action and Holography on Quasi-Fuchsian Deformation Spaces. Commun. Math. Phys. 362, 717–758 (2018). https://doi.org/10.1007/s00220-018-3164-2

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  • DOI: https://doi.org/10.1007/s00220-018-3164-2

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