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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2310.17536 |
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| _version_ | 1866910765281181696 |
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| author | Taghavi, Behrad Naseh, Ali Allameh, Kuroush |
| author_facet | Taghavi, Behrad Naseh, Ali Allameh, Kuroush |
| contents | 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 $\mathfrak{S}_{g,n}(\boldsymbol{m})$ 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 $\boldsymbol{m}=(m_1,\dots,m_n)$. 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 $\mathscr{L}_i$ over $\mathfrak{S}_{g,n}(\boldsymbol{m})$. These metrics are expressed in terms of the first coefficient of the expansion of covering map $J$ of 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 $\exp{S_{\boldsymbol{m}}/π}$ serves as a Hermitian metric on the $\mathbb{Q}$-line bundle $\mathscr{L}=\bigotimes_{i=1}^{n}\mathscr{L}_i^{\otimes (1-1/m_i^2)}$ over $\mathfrak{S}_{g,n}(\boldsymbol{m})$. Furthermore, we explicitly compute the first and second variations of the smooth real-valued function $\mathscr{S}_{\boldsymbol{m}}=S_{\boldsymbol{m}}-π\sum_{i=1}^n(m_i-\tfrac{1}{m_i})\log\mathsf{h}_{i}$ 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 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}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_17536 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Classical Liouville Action and Uniformization of Orbifold Riemann Surfaces Taghavi, Behrad Naseh, Ali Allameh, Kuroush High Energy Physics - Theory Mathematical Physics 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 $\mathfrak{S}_{g,n}(\boldsymbol{m})$ 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 $\boldsymbol{m}=(m_1,\dots,m_n)$. 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 $\mathscr{L}_i$ over $\mathfrak{S}_{g,n}(\boldsymbol{m})$. These metrics are expressed in terms of the first coefficient of the expansion of covering map $J$ of 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 $\exp{S_{\boldsymbol{m}}/π}$ serves as a Hermitian metric on the $\mathbb{Q}$-line bundle $\mathscr{L}=\bigotimes_{i=1}^{n}\mathscr{L}_i^{\otimes (1-1/m_i^2)}$ over $\mathfrak{S}_{g,n}(\boldsymbol{m})$. Furthermore, we explicitly compute the first and second variations of the smooth real-valued function $\mathscr{S}_{\boldsymbol{m}}=S_{\boldsymbol{m}}-π\sum_{i=1}^n(m_i-\tfrac{1}{m_i})\log\mathsf{h}_{i}$ 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 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}. |
| title | Classical Liouville Action and Uniformization of Orbifold Riemann Surfaces |
| topic | High Energy Physics - Theory Mathematical Physics |
| url | https://arxiv.org/abs/2310.17536 |