Counting zero-dimensional subschemes in higher dimensions

Yalong Cao; Martijn Kool

doi:10.1016/j.geomphys.2018.11.004

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arXiv · 1805.04746

Counting zero-dimensional subschemes in higher dimensions

Yalong Cao·Martijn Kool

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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.

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Yalong Cao, Martijn Kool. 2018-05-12. Counting zero-dimensional subschemes in higher dimensions. https://doi.org/10.1016/j.geomphys.2018.11.004

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