---
date: 2022-05-03 13:47
modification date: Tuesday 3rd May 2022 13:47:07
title: "instanton"
aliases: [instanton]
created: 2023-03-31T15:46
updated: 2023-03-31T16:00
---

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---

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# instanton

- A **Yang-Mills instanton** is a self-dual or [ASD](anti%20self%20dual.md) connection in a [principal bundle](torsor.md) over a four-dimensional [riemannian manifold](riemannian%20manifold.md) that plays the role of physical space-time in non-abelian [gauge theory](gauge%20theory.md). 
- Instantons are topologically nontrivial solutions of [Yang-Mills](Yang-Mills.md) equations that absolutely minimize the energy functional (within their topological type).

In $\Spin_7$ manifolds; why are these related to [Calabi-Yau fourfolds](Unsorted/Calabi-Yau%20fourfold.md)?

![](attachments/2023-03-31-7.png)
![](attachments/2023-03-31-8.png)

A $\operatorname{Spin}(7)$-instanton on $P$ is a connection $\nabla_P$ on $P$ with $\pi_7^2\left(F^{\nabla P}\right)=0$ in $\Gamma^{\infty}\left(\operatorname{Ad}(P) \otimes \Lambda_7^2 T^* X\right)$. 

Write $\mathcal{M}_P^{\mathrm{Spin}(7)}$ for the moduli space of irreducible Spin(7)-instantons on $P$. Then $\mathcal{M}_P^{\text {Spin( }(7)}$ is a derived manifold.