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 $d\not\equiv 2\mathrm{mod}4$. 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.

arXiv comment18 pages. Published version

References in corpus (6)

• Chern Classes of Tautological Sheaves on Hilbert Schemes
• Virtual fundamental classes for moduli spaces of sheaves on Calabi-Yau four-folds
• Zero-dimensional Donaldson-Thomas invariants of Calabi-Yau 4-folds
• Donaldson-Thomas theory for Calabi-Yau 4-folds
• Orientability for gauge theories on Calabi-Yau manifolds
• Magnificent Four

Cited by in corpus (4)

• Zero-dimensional Donaldson-Thomas invariants of Calabi-Yau 4-folds
• Canonical vertex formalism in DT theory of toric Calabi-Yau 4-folds
• Tetrahedron instantons
• Charging solid partitions