Renormalized volume, Polyakov anomaly, and orbifold Riemann surfaces

In [B. Taghavi Classical Liouville action and uniformization of orbifold Riemann surfaces, ], two of the authors studied the function ????m=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 ????m 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 ????m is equivalent to the Polyakov anomaly, which indicates that the function ????m 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, .].