Proof of a magnificent conjecture
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918243826925568 |
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| author | Kool, M. Rennemo, J. V. |
| author_facet | Kool, M. Rennemo, J. V. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_02852 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Proof of a magnificent conjecture Kool, M. Rennemo, J. V. Algebraic Geometry High Energy Physics - Theory Combinatorics 14C05, 14D21, 14N10, 14N35 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. |
| title | Proof of a magnificent conjecture |
| topic | Algebraic Geometry High Energy Physics - Theory Combinatorics 14C05, 14D21, 14N10, 14N35 |
| url | https://arxiv.org/abs/2507.02852 |