Renormalized Volume, Polyakov Anomaly and Orbifold Riemann Surfaces

TL;DR

This paper establishes a holographic duality between a specific function on orbifold Riemann surfaces and the renormalized hyperbolic volume of corresponding Schottky 3-orbifolds, linking geometric analysis with conformal field theory concepts.

Contribution

It proves the holographic duality between the function S_m and the renormalized hyperbolic volume for orbifold Riemann surfaces, extending previous classical Liouville action results.

Findings

V_{ren} acts as a Kähler potential for certain metrics.

S_m exhibits a Polyakov anomaly under conformal transformations.

The method offers an alternative derivation of the Polyakov anomaly for punctured Riemann surfaces.

Abstract

In arXiv:2310.17536, two of the authors studied the function $\mathscr{S}_{\boldsymbol{m}} = S_{\boldsymbol{m}} - \pi \sum_{i=1}^n (m_i - \tfrac{1}{m_i}) \log \mathsf{h}_{i}$ 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 K\"ahler 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.