The wheel classes in the locally finite homology of GLn(ℤ), canonical integrals and zeta values

Brown, Francis; Schnetz, Oliver

doi:10.2140/gt.2025.29.4389

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ISSN (electronic): 1364-0380

ISSN (print): 1465-3060

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The wheel classes in the locally finite homology of $\mathrm{GL}_n(\mathbb{Z})$, canonical integrals and zeta values

Francis Brown and Oliver Schnetz

Geometry & Topology 29 (2025) 4389–4447

DOI: 10.2140/gt.2025.29.4389

Abstract

We compute the canonical integrals associated to wheel graphs, and prove that they are proportional to odd zeta values. From this we deduce that wheel classes define explicit nonzero classes in: the locally finite homology of the general linear group ${GL}_{n} (ℤ)$ in both odd and even ranks, the homology of the moduli space of tropical curves, and the moduli space of tropical abelian varieties. We deduce the canonical integrals of a doubly infinite family of auxiliary classes in the even commutative graph complex. The proof also leads to a formula for the complete antisymmetrisation of a product of $2 n - 1$ matrices of rank $n$ , in the spirit of the Amitsur–Levitzki theorem.

Keywords

wheel integrals, Feynman integrals, wheel classes, graph homology, graph complex, general linear group, zeta values, periods, canonical forms

Mathematical Subject Classification

Primary: 11F06, 11F75, 11H55, 14T20, 20J06

Secondary: 11M32, 18G85, 81Q30

Supplementary material

Extra procedures for calculating canonical forms and integrals

References

Publication

Received: 16 March 2024

Revised: 13 January 2025

Accepted: 14 March 2025

Published: 26 November 2025

Proposed: Benson Farb

Seconded: Nathalie Wahl, Martin R Bridson

Authors

Francis Brown
All Souls College Oxford University Oxford United Kingdom

Oliver Schnetz
Institute for Theoretical Physics Hamburg Germany

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Volume 29, issue 8 (2025)