Classical Liouville Action and Uniformization of Orbifold Riemann Surfaces

Abstract

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 consider the generalized Schottky space ${\mathfrak{S}}_{g,n}(\mathbf{m})$ obtained as a holomorphic fibration over the Schottky space ${\mathfrak{S}}_g$ of the (compactified) underlying Riemann surface. Those fibers correspond to configuration spaces of $n$ orbifold points of orders $\mathbf{m}=(m_1,\dots ,m_n)$. Drawing on the previous work of Park, Teo, and Takhtajan \cite{park2015potentials} as well as Takhtajan and Zograf \cite{ZT_2018}, we define Hermitian metrics ${\mathsf{h}}_i$ for tautological line bundles ${\mathcal{L}}_i$ over ${\mathfrak{S}}_{g,n}(\mathbf{m})$. These metrics are expressed in terms of the first coefficient of the expansion of covering map $J$ of the Schottky domain. Additionally, we define the regularized classical Liouville action $S_{\mathbf{m}}$ using Schottky global coordinates on Riemann orbisurfaces with genus $g>1$. We demonstrate that $\exp S_{\mathbf{m}}/\pi$ serves as a Hermitian metric on the $\mathbb{Q}$-line bundle $\mathcal{L}={⨂}_{i=1}^n{\mathcal{L}}_i^{\otimes (1-1/m_i^2)}$ over ${\mathfrak{S}}_{g,n}(\mathbf{m})$. Furthermore, we explicitly compute the first and second variations of the smooth real-valued function ${\mathcal{S}}_{\mathbf{m}}=S_{\mathbf{m}}-\pi {\sum }_{i=1}^n(m_i-\frac{1}{m_i})\log {\mathsf{h}}_i$ on the Schottky deformation space ${\mathfrak{S}}_{g,n}(\mathbf{m})$. We establish two key results: (i) ${\mathcal{S}}_{\mathbf{m}}$ generates a combination of accessory and auxiliary parameters, and (ii) $-{\mathcal{S}}_{\mathbf{m}}$ 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 \cite{ZT_2018}.

arXiv comment95 pages + 66 pages appendix, 12 figures, added "Related works" in introduction and "Remark 5.1." about auxiliary parameters, typos corrected, references added

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