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Multiplicative convolution and double shuffle relations
[Submitted on 29 Apr 2026 (v1), last revised 22 Jun 2026 (this v · 2026-06-24 · via math updates on arXiv.org

Mathematics > Algebraic Geometry

arXiv:2604.26357 (math)

[Submitted on 29 Apr 2026 (v1), last revised 22 Jun 2026 (this version, v3)]

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Abstract: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.
Comments: v3: 32 pages, minor corrections, added an appendix on the homological definition of the self-intersection map. The first part of this paper previously appeared as arXiv:2412.15694
Subjects: Algebraic Geometry (math.AG); Algebraic Topology (math.AT); Number Theory (math.NT)
Report number: MPIM-Bonn-2026
Cite as: arXiv:2604.26357 [math.AG]
  (or arXiv:2604.26357v3 [math.AG] for this version)
  https://doi.org/10.48550/arXiv.2604.26357

arXiv-issued DOI via DataCite

Submission history

From: Nikita Markarian [view email]
[v1] Wed, 29 Apr 2026 07:08:52 UTC (45 KB)
[v2] Sat, 9 May 2026 07:11:58 UTC (45 KB)
[v3] Mon, 22 Jun 2026 19:16:21 UTC (49 KB)

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