Proof of a magnificent conjecture

TL;DR

This paper proves a conjecture related to weighted generating functions of solid partitions inspired by super-Yang-Mills theory, using advanced K-theoretic and geometric techniques to connect 4-dimensional invariants to 3-dimensional analogs.

Contribution

It introduces a novel geometric framework for K-theoretic invariants of Quot schemes and proves the Nekrasov-Piazzalunga conjecture by relating 4D invariants to 3D cases.

Findings

Recovered Nekrasov-Piazzalunga weights via localization

Derived sign rules for weighted generating functions

Connected 4D invariants to 3D analogs through limits

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.