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

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.

27 pages. Published version

References in corpus (5)

• Zero dimensional Donaldson-Thomas invariants of threefolds
• Virtual fundamental classes for moduli spaces of sheaves on Calabi-Yau four-folds
• Orientability for gauge theories on Calabi-Yau manifolds
• Counting zero-dimensional subschemes in higher dimensions
• Magnificent Four

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