Magnificent Four

Abstract

We present a statistical mechanical model whose random variables are solid partitions, i.e. Young diagrams built by stacking up four dimensional hypercubes. Equivalently, it can be viewed as the model of random tessellations of ${\bf R}^{3}$ by squashed cubes of four fixed orientations. The model computes the refined index of a system of D0-branes in the presence of $D8$-$\overline{D8}$ system, with a $B$-field strong enough to support the bound states. Mathematically, it is the equivariant K-theoretic version of integration over the Hilbert scheme of points on ${\bf C}^{4}$ and its higher rank analogues, albeit the definition is real, not complex analytic. The model is a mother of all random partition models, including the equivariant Donaldson-Thomas theory and the four dimensional instanton counting. Finally, a version of our model with infinite solid partitions with four fixed plane partition asymptotics is the vertex contribution to the equivariant count of instantons on toric Calabi-Yau fourfolds. The conjectured partition function of the model is presented. We have checked it up to six instantons (which is one step beyond the checks of the celebrated P.~MacMahon's failed conjectures of the early XX century). A specialization of the formula is our earlier (2004) conjecture on the equivariant K-theoretic Donaldson-Thomas theory, proven by A.~Okounkov in 2015.