$c_2$ invariants of hourglass chains via quadratic denominator reduction

TL;DR

This paper studies the $c_2$ invariants of hourglass chain graphs, deriving a formula based on the kernel, and finds connections to Calabi-Yau manifolds and modular forms, supporting the absence of curves in certain quantum field theory invariants.

Contribution

It introduces a formula for $c_2$ invariants of hourglass chains depending on the kernel and explores their relation to modular forms and Calabi-Yau manifolds.

Findings

$c_2$ invariants depend on the kernel of the graph.

Hourglass chains yield Calabi-Yau point-counts matching modular form coefficients.

No modular form of weight 2 and level ≤1000 matches the $c_2$ invariants.

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.