Communications in Number Theory and Physics

Volume 15 (2021)

Number 4

Geometries in perturbative quantum field theory

Pages: 743 – 791



Oliver Schnetz (Department Mathematik, Friedrich-Alexander-Universität Erlengen-Nürnberg, Erlangen, Germany)


In perturbative quantum field theory one encounters certain, very specific geometries over the integers. These perturbative quantum geometries determine the number contents of the amplitude considered. In the article ‘Modular forms in quantum field theory’ F. Brown and the author report on a first list of perturbative quantum geometries using the $c_2$-invariant in $\varphi^4$ theory. A main tool was denominator reduction which allowed the authors to examine graphs up to loop order (first Betti number) 10.

We introduce an improved quadratic denominator reduction which makes it possible to extend the previous results to loop order 11 (and partially orders 12 and 13). For comparison, also $\varphi^4$ graphs are investigated. Here, we extend the results from loop order 9 to 10. The new database of 4801 unique $c_2$-invariants (previously 157)—while being consistent with all major $c_2$-conjectures—leads to a more refined picture of perturbative quantum geometries. In the appendix, Friedrich Knop proves a Chevalley–Warning–Ax theorem for double covers of affine space.


Feynman period, $c_2$-invariant

2010 Mathematics Subject Classification

14M99, 81T99

Appendix by Friedrich Knop (Department Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen, Germany)

Oliver Schnetz is supported by the DFG grant SCHN 1240.

Received 17 May 2019

Accepted 9 June 2021

Published 6 October 2021