𝑐₂ Invariants of Hourglass Chains via Quadratic Denominator Reduction

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 𝑐₂ invariant of these hourglass chains, which only depends on the kernel. For different kernels, these hourglass chains typically give rise to different 𝑐₂ invariants. An exhaustive search for the 𝑐₂ 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 ≤ 1000 corresponds to the 𝑐₂ of such hourglass chains. This provides further evidence in favour of the conjecture that curves are absent in 𝑐₂ invariants of 𝜙⁴ quantum field theory.