Multiplicative convolution and double shuffle relations
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866910204429336576 |
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| author | Markarian, Nikita |
| author_facet | Markarian, Nikita |
| contents | We develop a geometric approach to the regularized double shuffle relations for multiple zeta values, based on convolution of perverse sheaves on $\mathbb{C}^*$ and inspired by the approach of Deligne and Terasoma. We introduce semi-holonomy isomorphisms associated with pro-unipotent paths and show that their compatibility with multiplicative convolution is equivalent to a condition on the pro-unipotent fundamental group, the homological pentagon equation. We prove that this condition is equivalent to the regularized double shuffle relations, yielding a geometric proof that the pentagon equation implies these relations. The approach is purely topological and avoids Hodge-theoretic and Tannakian methods. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_26357 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Multiplicative convolution and double shuffle relations Markarian, Nikita Algebraic Geometry Algebraic Topology Number Theory We develop a geometric approach to the regularized double shuffle relations for multiple zeta values, based on convolution of perverse sheaves on $\mathbb{C}^*$ and inspired by the approach of Deligne and Terasoma. We introduce semi-holonomy isomorphisms associated with pro-unipotent paths and show that their compatibility with multiplicative convolution is equivalent to a condition on the pro-unipotent fundamental group, the homological pentagon equation. We prove that this condition is equivalent to the regularized double shuffle relations, yielding a geometric proof that the pentagon equation implies these relations. The approach is purely topological and avoids Hodge-theoretic and Tannakian methods. |
| title | Multiplicative convolution and double shuffle relations |
| topic | Algebraic Geometry Algebraic Topology Number Theory |
| url | https://arxiv.org/abs/2604.26357 |