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 DT4 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.