Charging solid partitions

Solid partitions are the 4D generalization of the plane partitions in 3D and Young diagrams in 2D, and they can be visualized as stacking of 4D unit-size boxes in the positive corner of a 4D room. Physically, solid partitions arise naturally as 4D molten crystals that count equivariant D-brane BPS states on the simplest toric Calabi-Yau fourfold, $\mathbb{C}^4$, generalizing the 3D statement that plane partitions count equivariant D-brane BPS states on $\mathbb{C}^3$. In the construction of BPS algebras for toric Calabi-Yau threefolds, the so-called charge function on the 3D molten crystal is an important ingredient -- it is the generating function for the eigenvalues of an infinite tower of Cartan elements of the algebra. In this paper, we derive the charge function for solid partitions. Compared to the 3D case, the new feature is the appearance of contributions from certain 4-box and 5-box clusters, which will make the construction of the corresponding BPS algebra much more complicated than in the 3D.