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$c_2$ Invariants of Hourglass Chains via Quadratic Denominator Reduction
Oliver Schnetza, Karen Yeatsb 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
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.
Keywords:
$c_2$ invariant, denominator reduction, quadratic denominator reduction, Feynman period.
Received: February 25, 2021; in final form November 2, 2021; Published online November 10, 2021
Citation:
Oliver Schnetz, Karen Yeats, “$c_2$ Invariants of Hourglass Chains via Quadratic Denominator Reduction”, SIGMA, 17 (2021), 100, 26 pp.
Linking options:
https://www.mathnet.ru/eng/sigma1782 https://www.mathnet.ru/eng/sigma/v17/p100
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Abstract page: | 48 | Full-text PDF : | 12 | References: | 6 |
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