Classical Liouville action and uniformization of orbifold Riemann surfaces

We study the classical Liouville field theory on Riemann surfaces of genus $g>1$ in the presence of vertex operators associated with branch points of orders $m_i>1$. In order to do so, we will consider the generalized Schottky space $S_{g,n}\left(m\right)$ obtained as a holomorphic fibration over the Schottky space $S_g$ of the (compactified) underlying Riemann surface. The fibers of $S_{g,n}\left(m\right)\to S_g$ correspond to configuration spaces of $n$ orbifold points of orders $m=(m_1,\dots ,m_n)$. Drawing on the previous work of Park et al. [Adv. Math. 305, 856 (2017)] as well as Takhtajan and Zograf [Lett. Math. Phys. 109, 1119 (2018); L. A. Takhtajan and P. ZografLett. Math. Phys.114, 60 (2024)], we define Hermitian metrics $h_i$ for tautological line bundles ${ℒ}_i$ over $S_{g,n}\left(m\right)$. These metrics are expressed in terms of the first coefficient of the expansion of covering map $J$ near each singular point on the Schottky domain. Additionally, we define the regularized classical Liouville action $S_m$ using Schottky global coordinates on Riemann orbisurfaces with genus $g>1$. We demonstrate that $\exp [S_m/\pi ]$ serves as a Hermitian metric in the holomorphic $Q$-line bundle $ℒ={\otimes }_{i=1}^n{ℒ}_i^{\otimes (1-1/m_i^2)}$ over $S_{g,n}\left(m\right)$. Furthermore, we explicitly compute the first and second variations of the smooth real-valued function ${𝒮}_m=S_m-\pi {\sum }_{i=1}^n(m_i-\frac{1}{m_i})\log h_i$ on the Schottky deformation space $S_{g,n}\left(m\right)$. We establish two key results: (i) ${𝒮}_m$ generates a combination of accessory and auxiliary parameters, and (ii) $-{𝒮}_m$ acts as a Kähler potential for a specific combination of Weil-Petersson and Takhtajan-Zograf metrics that appear in the local index theorem for orbifold Riemann surfaces [Takhtajan and Zograf, Lett. Math. Phys. 109, 1119 (2018)]. The obtained results can then be interpreted in terms of the complex geometry of the Hodge line bundle equipped with Quillen's metric over the moduli space $M_{g,n}\left(m\right)$ of Riemann orbisurfaces and the tree-level approximation of conformal Ward identities associated with quantum Liouville theory.