Communications in Number Theory and Physics

Volume 17 (2023)

Number 2

Completing the $c_2$ completion conjecture for $p=2$

Pages: 343 – 384

DOI: https://dx.doi.org/10.4310/CNTP.2023.v17.n2.a4

Authors

Simone Hu (Mathematical Institute, University of Oxford, United Kingdom)

Karen Yeats (Department of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, Ontario, Canada)

Abstract

The $c_2$-invariant is an arithmetic graph invariant useful for understanding Feynman periods. Brown and Schnetz conjectured that the $c_2$-invariant has a particular symmetry known as completion invariance. This paper will prove completion invariance of the $c_2$-invariant in the $p=2$ case, extending previous work of one of us. The methods are combinatorial and enumerative involving counting certain partitions of the edges of the graph.

Keywords

Feynman period, completion, $c_2$-invariant, edge partition

2010 Mathematics Subject Classification

Primary 81T18. Secondary 05C30, 05C31, 81Q30.

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K.Y. is supported by an NSERC Discovery grant and by the Canada Research Chairs program.

Received 23 June 2022

Accepted 22 March 2023

Published 4 May 2023