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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2402.12529 |
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| _version_ | 1866917098953900032 |
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| author | Kokotov, Alexey Korikov, Dmitrii |
| author_facet | Kokotov, Alexey Korikov, Dmitrii |
| contents | We study the regularized determinants ${\rm det}\, Δ$ of various self-adjoint extensions of symmetric Laplacians acting in spinor bundles over compact Riemann surfaces with flat singular metrics $|ω|^2$, where $ω$ is a holomorphic one form on the Riemann surface. We find an explicit expression for ${\rm det}\, Δ$ for the so-called self-adjoint Szegö extension through the Bergman tau-function on the moduli space of Abelian differentials and the theta-constants (corresponding to the spinor bundle). This expression can be considered as a version of the well-known spin-$1/2$ bosonization formula of Bost-Nelson for the case of flat conformal metrics with conical singularities and a higher genus generalization of the Ray-Singer formula for flat elliptic curves. We establish comparison formulas for the determinants of two different extensions (e. g., the Szegö extension and the Friedrichs one). The paper answers a question raised by D'Hoker and Phong \cite{DH-P} more than thirty years ago. We also reconsider the results from \cite{DH-P} on the regularization of diverging determinant ratio for Mandelstam metrics (for any spin) proposing (and computing) a new regularization of this ratio. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_12529 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Laplacians in spinor bundles over translation surfaces: self-adjoint extentions and regularized determinants Kokotov, Alexey Korikov, Dmitrii Differential Geometry Spectral Theory We study the regularized determinants ${\rm det}\, Δ$ of various self-adjoint extensions of symmetric Laplacians acting in spinor bundles over compact Riemann surfaces with flat singular metrics $|ω|^2$, where $ω$ is a holomorphic one form on the Riemann surface. We find an explicit expression for ${\rm det}\, Δ$ for the so-called self-adjoint Szegö extension through the Bergman tau-function on the moduli space of Abelian differentials and the theta-constants (corresponding to the spinor bundle). This expression can be considered as a version of the well-known spin-$1/2$ bosonization formula of Bost-Nelson for the case of flat conformal metrics with conical singularities and a higher genus generalization of the Ray-Singer formula for flat elliptic curves. We establish comparison formulas for the determinants of two different extensions (e. g., the Szegö extension and the Friedrichs one). The paper answers a question raised by D'Hoker and Phong \cite{DH-P} more than thirty years ago. We also reconsider the results from \cite{DH-P} on the regularization of diverging determinant ratio for Mandelstam metrics (for any spin) proposing (and computing) a new regularization of this ratio. |
| title | Laplacians in spinor bundles over translation surfaces: self-adjoint extentions and regularized determinants |
| topic | Differential Geometry Spectral Theory |
| url | https://arxiv.org/abs/2402.12529 |