“School of Particles And Accelerator”
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| Paper IPM / Particles And Accelerator / 17864 | ||||||||||||||||||||||
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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 \mathfrak{S}_{g,n}(\boldsymbol{m}) obtained as a holomorphic fibration over the Schottky space \mathfrak{S}_g of the (compactified) underlying Riemann surface. The fibers of \mathfrak{S}_{g,n}(\boldsymbol{m}) \to \mathfrak{S}_g correspond to configuration spaces of n orbifold points of orders \boldsymbol{m} = (m_1,\dots,m_n). Park et al. [\href{https://www.sciencedirect.com/science/article/pii/S0001870816301670?via%3Dihub}{Adv. Math. 305, 856 (2017)}] as well as Takhtajan and Zograf [\href{https://link.springer.com/article/10.1007/s11005-018-01144-w}{Lett. Math. Phys. 109, 1119 (2018)}; L.âA. Takhtajan and P. Zograf \href{https://link.springer.com/article/10.1007/s11005-024-01809-9}{Lett. Math. Phys.114, 60 (2024)}], we define Hermitian metrics \mathsf{h}_i for tautological line bundles \mathcal{L}_i over \mathfrak{S}_{g,n}(\boldsymbol{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 S_{\boldsymbol{m}} using Schottky global coordinates on Riemann orbisurfaces with genus g>1. We demonstrate that \exp[S_{\boldsymbol{m}}/\pi] serves as a Hermitian metric in the holomorphic \mathbb{Q}-line bundle \mathcal{L} = \bigotimes_{i=1}^{n} \mathcal{L}_i^{\otimes (1-1/m_i^2)} over \mathfrak{S}_{g,n}(\boldsymbol{m}). Furthermore, we explicitly compute the first and second variations of the smooth real-valued function \mathscr{S}_{\boldsymbol{m}} = S_{\boldsymbol{m}} - \pi \sum_{i=1}^n (m_i - \tfrac{1}{m_i}) \log \mathsf{h}_{i} on the Schottky deformation space \mathfrak{S}_{g,n}(\boldsymbol{m}). We establish two key results: (i) \mathscr{S}_{\boldsymbol{m}} generates a combination of accessory and auxiliary parameters, and (ii) -\mathscr{S}_{\boldsymbol{m}} acts as a K\"{a}hler 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, \href{https://link.springer.com/article/10.1007/s11005-018-01144-w}{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 \mathfrak{M}_{g,n}(\boldsymbol{m}) of Riemann orbisurfaces and the tree-level approximation of conformal Ward identities associated with quantum Liouville theory.
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