Counting zero-dimensional subschemes in higher dimensions

TL;DR

This paper explores zero-dimensional Donaldson-Thomas invariants across various dimensions, revealing special behaviors in dimensions 3 and 4, and proposing that certain conjectures hold universally after parameter specialization.

Contribution

It extends the study of Donaldson-Thomas invariants to higher dimensions, identifying dimension-specific phenomena and proposing a unifying approach through parameter specialization.

Findings

Invariants can be expressed via the MacMahon function in certain cases.

Dimensions 3 and 4 exhibit unique behaviors in these invariants.

Specializing parameters suggests conjectures hold in all 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.