Constructing polylogarithms on higher-genus Riemann surfaces
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866909530525270016 |
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| author | D'Hoker, Eric Hidding, Martijn Schlotterer, Oliver |
| author_facet | D'Hoker, Eric Hidding, Martijn Schlotterer, Oliver |
| contents | An explicit construction is presented of homotopy-invariant iterated integrals on a Riemann surface of arbitrary genus in terms of a flat connection valued in a freely generated Lie algebra. The integration kernels consist of modular tensors, built from convolutions of the Arakelov Green function and its derivatives with holomorphic Abelian differentials, combined into a flat connection. Our construction thereby produces explicit formulas for polylogarithms as higher-genus modular tensors. This construction generalizes the elliptic polylogarithms of Brown-Levin, and prompts future investigations into the relation with the function spaces of higher-genus polylogarithms in the work of Enriquez-Zerbini. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_08644 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Constructing polylogarithms on higher-genus Riemann surfaces D'Hoker, Eric Hidding, Martijn Schlotterer, Oliver High Energy Physics - Theory Algebraic Geometry Number Theory An explicit construction is presented of homotopy-invariant iterated integrals on a Riemann surface of arbitrary genus in terms of a flat connection valued in a freely generated Lie algebra. The integration kernels consist of modular tensors, built from convolutions of the Arakelov Green function and its derivatives with holomorphic Abelian differentials, combined into a flat connection. Our construction thereby produces explicit formulas for polylogarithms as higher-genus modular tensors. This construction generalizes the elliptic polylogarithms of Brown-Levin, and prompts future investigations into the relation with the function spaces of higher-genus polylogarithms in the work of Enriquez-Zerbini. |
| title | Constructing polylogarithms on higher-genus Riemann surfaces |
| topic | High Energy Physics - Theory Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2306.08644 |