“School of Particles”
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| Paper IPM / Particles / 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 $m_i>1$. 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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