WWDC26：2026 年 6 月 8 日至 12 日 (太平洋时间)

We consider a hyperkähler $8$-manifold admitting either a $U(1) \times \mathbb{R}$, or a $U(1) \times U(1)$ action, where the first factor preserves $g$ and $I$, and acts on $ω_2+iω_3$ by multiplying it by itself, while the second factor preserves $g$ and acts triholomorphically. Such data can be reduced to a single function $H$ of two complex variables and two real variables satisfying $6$ equations of Monge-Ampere type, which can be compactly written down using a Poisson bracket.

We give a complete characterization of the parity of $b_8(n)$, the number of $8$-regular partitions of $n$. Namely, we prove that $b_8(n)$ is odd or even depending on whether or not we have the factorisation $24n+7=p^{4a+1}m^2$, for some prime $p\nmid m$ and $a\ge 0$.