Virtual invariants from the non-associative Hilbert scheme

We introduce a non-associative model for the Hilbert scheme of points in arbitrary dimension. We define a smooth ambient space, which we call the non-associative Hilbert scheme, containing the classical nested Hilbert scheme $\mathrm{NHilb}^{\underline{d}}(\mathbb{A}^n)$ as the associativity, cut out by an explicit section of an associativity bundle. This construction yields canonical perfect obstruction theories and virtual fundamental classes on $\mathrm{NHilb}^{\underline{d}}(\mathbb{A}^n)$ for all $(n,\underline d)$. Using virtual localization, we obtain closed formulas for these virtual classes as sums over admissible nested partitions. Over the punctual locus, we rewrite these as a single multivariable iterated residue formula governing all virtual integrals. Our construction works for all $n$, produces positive-dimensional virtual classes when $n$ is large compared to the number of points, and we expect that they extend the non-commutative matrix model and virtual class construction on Calabi-Yau threefolds.

💡 Research Summary

The authors introduce a novel “non‑associative Hilbert scheme” as a smooth ambient space that contains the classical nested Hilbert scheme of points on affine space Aⁿ as the locus where associativity holds. Starting from the free unital non‑associative algebra C{ x₁,…,xₙ }ᶜ, they define the moduli space naNHilbᵈ(Aⁿ) of chains of ideals in this algebra. This space is smooth for any dimension vector d=(d₀,…,dᵣ) and any n. The key observation is that the associativity constraints can be encoded as a global section s_ass of a vector bundle E_ass over naNHilbᵈ(Aⁿ). The zero‑locus of s_ass is precisely the classical nested Hilbert scheme NHilbᵈ(Aⁿ). Consequently, NHilbᵈ(Aⁿ) inherits a canonical perfect obstruction theory (in the sense of Behrend–Fantechi) and thus a virtual fundamental class