The local moduli space of the Einstein-Yang-Mills system
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| author | Bunk, Severin Muñoz, Vicente Shahbazi, C. S. |
| author_facet | Bunk, Severin Muñoz, Vicente Shahbazi, C. S. |
| contents | We study the deformation theory of the Einstein-Yang-Mills system on a principal bundle with a compact structure group over a compact manifold. We first construct, as an application of the general slice theorem of Diez and Rudolph, a smooth slice in the tame Fréchet category for the coupled action of bundle automorphisms on metrics and connections. Using this result, together with a careful analysis of the linearization of the Einstein-Yang-Mills system, we realize the moduli space of Einstein-Yang-Mills pairs modulo automorphism as an analytic set in a finite-dimensional tame Fréchet manifold, extending classical results of Koiso for Einstein metrics and Yang-Mills connections to the Einstein-Yang-Mills system. Furthermore, we introduce the notion of \emph{essential deformation} of an Einstein-Yang-Mills pair, which we characterize in full generality and explore in more detail in the four-dimensional case, proving a decoupling result for trace deformations when the underlying Einstein-Yang-Mills pair is a Ricci-flat metric coupled to an anti-self-dual instanton. In particular, we find a novel obstruction that does not occur in the \emph{decoupled} Einstein or Yang-Mills moduli problems. Finally, we prove that every essential deformation of the four-dimensional Einstein-Yang-Mills system based on a Calabi-Yau metric coupled to an instanton is of restricted type. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_07572 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The local moduli space of the Einstein-Yang-Mills system Bunk, Severin Muñoz, Vicente Shahbazi, C. S. Differential Geometry We study the deformation theory of the Einstein-Yang-Mills system on a principal bundle with a compact structure group over a compact manifold. We first construct, as an application of the general slice theorem of Diez and Rudolph, a smooth slice in the tame Fréchet category for the coupled action of bundle automorphisms on metrics and connections. Using this result, together with a careful analysis of the linearization of the Einstein-Yang-Mills system, we realize the moduli space of Einstein-Yang-Mills pairs modulo automorphism as an analytic set in a finite-dimensional tame Fréchet manifold, extending classical results of Koiso for Einstein metrics and Yang-Mills connections to the Einstein-Yang-Mills system. Furthermore, we introduce the notion of \emph{essential deformation} of an Einstein-Yang-Mills pair, which we characterize in full generality and explore in more detail in the four-dimensional case, proving a decoupling result for trace deformations when the underlying Einstein-Yang-Mills pair is a Ricci-flat metric coupled to an anti-self-dual instanton. In particular, we find a novel obstruction that does not occur in the \emph{decoupled} Einstein or Yang-Mills moduli problems. Finally, we prove that every essential deformation of the four-dimensional Einstein-Yang-Mills system based on a Calabi-Yau metric coupled to an instanton is of restricted type. |
| title | The local moduli space of the Einstein-Yang-Mills system |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2311.07572 |