Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 100, 26 pages      arXiv:2102.12383      https://doi.org/10.3842/SIGMA.2021.100
Contribution to the Special Issue on Algebraic Structures in Perturbative Quantum Field Theory in honor of Dirk Kreimer for his 60th birthday

$c_2$ Invariants of Hourglass Chains via Quadratic Denominator Reduction

Oliver Schnetz a and Karen Yeats b
a) Department Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstrasse 11, 91058, Erlangen, Germany
b) Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada

Received February 25, 2021, in final form November 02, 2021; Published online November 10, 2021

Abstract
We introduce families of four-regular graphs consisting of chains of hourglasses which are attached to a finite kernel. We prove a formula for the $c_2$ invariant of these hourglass chains which only depends on the kernel. For different kernels these hourglass chains typically give rise to different $c_2$ invariants. An exhaustive search for the $c_2$ invariants of hourglass chains with kernels that have a maximum of ten vertices provides Calabi-Yau manifolds with point-counts which match the Fourier coefficients of modular forms whose weights and levels are [4,8], [4,16], [6,4], and [9,4]. Assuming the completion conjecture, we show that no modular form of weight 2 and level $\leq1000$ corresponds to the $c_2$ of such hourglass chains. This provides further evidence in favour of the conjecture that curves are absent in $c_2$ invariants of $\phi^4$ quantum field theory.

Key words: $c_2$ invariant; denominator reduction; quadratic denominator reduction; Feynman period.

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