Proof of a magnificent conjecture

Abstract

Motivated by super-Yang-Mills theory on a Calabi-Yau 4-fold, Nekrasov and Piazzalunga have assigned weights to $r$-tuples of solid partitions and conjectured a formula for their weighted generating function. We define $K$-theoretic virtual invariants of Quot schemes of 0-dimensional quotients of $\mathcal{O}_{\mathbb{C}^4}^{\oplus r}$ by realizing them as zero loci of isotropic sections of orthogonal bundles on non-commutative Quot schemes. Via the Oh-Thomas localization formula, we recover Nekrasov-Piazzalunga's weights and derive their sign rule. Our proof passes through refining the $K$-theoretic invariants to sheaves and describing them via Clifford modules, which lets us show that they arise from a factorizable sequence of sheaves in the sense of Okounkov. Taking limits of the equivariant parameters, we then deduce the Nekrasov-Piazzalunga conjecture from its 3-dimensional analog.