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 mi>1. In order to do so, we will consider the generalized Schottky space Sg,n(m) obtained as a holomorphic fibration over the Schottky space Sg of the (compactified) underlying Riemann surface. The fibers of Sg,n(m)→Sg correspond to configuration spaces of n orbifold points of orders m=(m1,…,mn). Drawing on the previous work of Park as well as Takhtajan and Zograf ; L. A. Takhtajan and P. Zograf, we define Hermitian metrics hi for tautological line bundles ℒi over Sg,n(m). 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 Sm using Schottky global coordinates on Riemann orbisurfaces with genus g>1. We demonstrate that expSm/π serves as a Hermitian metric in the holomorphic Q-line bundle ℒ=⊗i=1nℒi⊗(1−1/mi2) over Sg,n(m). Furthermore, we explicitly compute the first and second variations of the smooth real-valued function 𝒮m=Sm−π∑i=1n(mi−1mi)loghi on the Schottky deformation space Sg,n(m). We establish two key results: (i) 𝒮m generates a combination of accessory and auxiliary parameters, and (ii) −𝒮m 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, . 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 Mg,n(m) of Riemann orbisurfaces and the tree-level approximation of conformal Ward identities associated with quantum Liouville theory. Published by the American Physical Society 2024