Counting zero-dimensional subschemes in higher dimensions

Abstract

Consider zero-dimensional Donaldson–Thomas invariants of a toric threefold or toric Calabi–Yau fourfold. In the second case, invariants can be defined using a tautological insertion. In both cases, the generating series can be expressed in terms of the MacMahon function. In the first case, this follows from a theorem of Maulik–Nekrasov–Okounkov–Pandharipande. In the second case, this follows from a conjecture of the authors and a (more general K-theoretic) conjecture of Nekrasov.In this paper, we consider formal analogues of these invariants in any dimension d⁄≡2mod4. The direct analogues of the above-mentioned conjectures fail in general when d>4, showing that dimensions 3 and 4 are special. Surprisingly, after appropriate specialization of the equivariant parameters, the conjectures seem to hold in all dimensions.