Zero-dimensional Donaldson-Thomas invariants of Calabi-Yau 4-folds

TL;DR

This paper introduces and studies zero-dimensional Donaldson-Thomas invariants for Calabi-Yau 4-folds, proposing conjectures and verifying them in specific cases, and connects these invariants to solid partitions and the MacMahon function.

Contribution

It defines new DT4 invariants for Calabi-Yau 4-folds, formulates conjectures for their generating series, and verifies these in cases involving toric divisors and solid partitions.

Findings

Conjectured generating series formula for DT4 invariants.

Verified conjecture for smooth toric divisors.

Connected weighted solid partition generating function to the MacMahon function.

Abstract

We study Hilbert schemes of points on a smooth projective Calabi-Yau 4-fold $X$. We define $\mathrm{DT}_4$ invariants by integrating the Euler class of a tautological vector bundle $L^{[n]}$ against the virtual class. We conjecture a formula for their generating series, which we prove in certain cases when $L$ corresponds to a smooth divisor on $X$. A parallel equivariant conjecture for toric Calabi-Yau 4-folds is proposed. This conjecture is proved for smooth toric divisors and verified for more general toric divisors in many examples. Combining the equivariant conjecture with a vertex calculation, we find explicit positive rational weights, which can be assigned to solid partitions. The weighted generating function of solid partitions is given by $\exp(M(q)-1)$, where $M(q)$ denotes the MacMahon function.