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Pentagram relations
Abstract
We study the cohomology of iterated Kummer extensions on the moduli space of five-pointed rational curves, continuing [arXiv:2609.19341, arXiv:2609.21356]. A two-dimensional shrinking construction compares the Betti presentations obtained from its five forgetful projections. The corresponding Hodge defects give pentagram relations, quadratic relations between coefficients of the Drinfeld associator, which are equivalent to the pentagon equation under group-likeness, normalization, and reversal. Under reversal and vanishing of pure-power coefficients, the relations imply shuffle and regularized double shuffle, whose Hodge realization is recovered by a convolution specialization.
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BibTeX
@misc{markarian2026pentagram,
title = {{Pentagram relations}},
author = {Nikita Markarian},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00225},
eprint = {2610.00225},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00225},
}
RIS
TY - UNPB
TI - Pentagram relations
AU - Nikita Markarian
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00225
UR - https://arxiv.org/abs/2610.00225
AB - We study the cohomology of iterated Kummer extensions on the moduli space of five-pointed rational curves, continuing [arXiv:2609.19341, arXiv:2609.21356]. A two-dimensional shrinking construction compares the Betti presentations obtained from its five forgetful projections. The corresponding Hodge defects give pentagram relations, quadratic relations between coefficients of the Drinfeld associator, which are equivalent to the pentagon equation under group-likeness, normalization, and reversal. Under reversal and vanishing of pure-power coefficients, the relations imply shuffle and regularized double shuffle, whose Hodge realization is recovered by a convolution specialization.
ER -