Geometries in perturbative quantum field theory

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 $\phi^4$ 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 'non-$\phi^4$' graphs are investigated. Here, we were able to 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.Comment: 35 page

Some identities for enumerators of circulant graphs

We establish analytically several new identities connecting enumerators of different types of circulant graphs of prime, twice prime and prime-squared orders. In particular, it is shown that the semi-sum of the number of undirected circulants and the number of undirected self-complementary circulants of prime order is equal to the number of directed self-complementary circulants of the same order. Keywords: circulant graph; cycle index; cyclic group; nearly doubled primes; Cunningham chain; self-complementary graph; tournament; mixed graphComment: 17 pages, 3 tables Categories: CO Combinatorics (NT Number Theory) Math Subject Class: 05C30; 05A19; 11A4