In [B. Taghavi Classical Liouville action and uniformization of orbifold Riemann surfaces, ], two of the authors studied the function 𝒮m=Sm−π∑i=1n(mi−1mi)loghi for orbifold Riemann surfaces of signature (g;m1,…,mne;np) on the generalized Schottky space Sg,n(m). In this paper, we prove the holographic duality between 𝒮m and the renormalized hyperbolic volume Vren of the corresponding Schottky 3-orbifolds with lines of conical singularity that reach the conformal boundary. In case of the classical Liouville action on Sg and Sg,n(∞), the holography principle was proved in [K. Krasnov, Holography and Riemann surfaces, , 929 (2000).], [J. Park , Potentials and Chern forms for Weil–Petersson and Takhtajan–Zograf metrics on moduli spaces, , 856 (2017).], respectively. Our result implies that Vren acts as a Kähler potential for a particular combination of the Weil-Petersson and Takhtajan-Zograf metrics that appears in the local index theorem for orbifold Riemann surfaces, as discussed by [L. A. Takhtajan and P. Zograf, Local index theorem for orbifold Riemann surfaces, .]. Moreover, we demonstrate that under the conformal transformations, the change of function 𝒮m is equivalent to the Polyakov anomaly, which indicates that the function 𝒮m is a consistent height function with a unique hyperbolic solution. Consequently, the associated renormalized hyperbolic volume Vren also admits a Polyakov anomaly formula. The method we used to establish this equivalence may provide an alternative approach to derive the renormalized Polyakov anomaly for Riemann surfaces with punctures (cusps), as described by [P. Albin , Ricci flow and the determinant of the Laplacian on non-compact surfaces, .]. Published by the American Physical Society 2025

中文翻译：

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重整化体积、Polyakov 异常和环状黎曼曲面


在 [B. Taghavi 经典 Liouville 作用和球形黎曼曲面的均匀化，]中，两位作者π∑研究了特征 （g;m1,...,mne;np） 在广义肖特基空间 Sg，n（m） 上。在本文中，我们证明了 Sm 与相应的肖特基 3 球折叠的重新归一化双曲体积 Vren 之间的全息对偶性，其圆锥形奇点线到达共形边界。在经典的 Liouville 作用对 Sg 和 Sg，n（∞） 的情况下，全息原理分别在 [K. Krasnov， Holography and Riemann surfaces， ， 929 （2000）.]， [J. Park， Potentials and Chern forms for Weil-Petersson and Takhtajan-Zograf metrics on moduli spaces， ， 856 （2017）.] 中得到证明。我们的结果表明，Vren 充当 Weil-Petersson 和 Takhtajan-Zograf 度量的特定组合的 Kähler 势，该度量出现在球形黎曼曲面的局部索引定理中，如 [L. A. Takhtajan 和 P. Zograf， Orbifold Riemann 曲面的局部索引定理，.] 所讨论的那样。此外，我们证明了在共形变换下，函数 Sm 的变化等同于 Polyakov 异常，这表明函数 Sm 是一个具有独特双曲解的一致高度函数。因此，相关的重整化双曲体积 Vren 也承认 Polyakov 异常公式。我们用来建立这种等价性的方法可能提供一种替代方法来推导出具有穿刺（尖点）的黎曼曲面的重整化 Polyakov 异常，如 [P. Albin， Ricci flow and the deterfactor of the Laplaceian on non-compact surfaces， .] 所述。 美国物理学会 2025 年出版