Renormalized Volume, Polyakov Anomaly and Orbifold Riemann Surfaces

TL;DR

This work proves a holographic duality between the generalized Liouville action $S_m$ on orbifold Schottky space and the renormalized hyperbolic volume $V_{ren}$ of Schottky 3‑orbifolds with boundary cusps and conical lines. Using a Γ‑invariant regularization surface, the authors show that $V_{ren}$ coincides (up to an area term) with $S_m$, thereby making $V_{ren}$ a Kaehler potential for a Weil‑Petersson–Takhtajan‑Zograf combination in the orbifold setting. They further establish a Polyakov anomaly for the renormalized quantities under conformal changes, extending the known compact‑surface results to orbifolds with punctures and conical points, and outline a method that may offer an alternative route to renormalized Polyakov anomalies for cusps. Overall, the paper extends holography for Liouville theory to orbifold Schottky data, connecting 3D gravity quantities to 2D conformal geometry and spectral data, with implications for quantum gravity and isospectrality phenomena. These results enrich the geometric understanding of holography, uniformization, and modular geometry in the presence of orbifold singularities.

Abstract

In arXiv:2310.17536, two of the authors studied the function $\mathscr{S}_{\boldsymbol{m}} = S_{\boldsymbol{m}} - π\sum_{i=1}^n (m_i - \tfrac{1}{m_i}) \log \mathsf{h}_{i}$ for orbifold Riemann surfaces of signature $(g;m_1,...,m_{n_e};n_p)$ on the generalized Schottky space $\mathfrak{S}_{g,n}(\boldsymbol{m})$. In this paper, we prove the holographic duality between $\mathscr{S}_{\boldsymbol{m}}$ and the renormalized hyperbolic volume $V_{\text{ren}}$ of the corresponding Schottky 3-orbifolds with lines of conical singularity that reach the conformal boundary. In case of the classical Liouville action on $\mathfrak{S}_{g}$ and $\mathfrak{S}_{g,n}(\boldsymbol{\infty})$, the holography principle was proved in arXiv:hep-th/0005106v2 and arXiv:1508.02102, respectively. Our result implies that $V_{\text{ren}}$ acts as 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 arXiv:1701.00771. Moreover, we demonstrate that under the conformal transformations, the change of function $\mathscr{S}_{\boldsymbol{m}}$ is equivalent to the Polyakov anomaly, which indicates that the function $\mathscr{S}_{\boldsymbol{m}}$ is a consistent height function with a unique hyperbolic solution. Consequently, the associated renormalized hyperbolic volume $V_{\text{ren}}$ 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 in arXiv:0909.0807.

Figures (5)

• Figure 1: Fundametnal domain $\mathcal{D}$ of Schottky uniformization for $g=2$.
• Figure 2: The three dimensional extension of fundamental domain $\mathcal{D}$ in Figure \ref{['fig:1']}.
• Figure 3: The three-dimensional extension of fundamental domain with defect points. The left figure is its fundamental domain $\overset{{}_{\curlywedge}}{\mathcal{D}}_{\delta}$, and the right figure shows the geodesic extension to $\mathbb{U}^3$.
• Figure 4: The regularizing surface $f(Z)$ which cuts through the fundamental region.
• Figure 5: A sketch of pulling back partition of unity from 2-orbifold to $\Omega$ by inverse projection map $\pi^{-1}$. The same idea holds for 3D bulk.

Theorems & Definitions (2)

• Lemma 1: I.kra kra1972
• proof