The Compactness Problem for Special Holonomy Manifolds
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2025
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| author | Revista, Zen MATH, 10 |
| author_facet | Revista, Zen MATH, 10 |
| contents | The study of special holonomy manifolds represents a vibrant and challenging area within differential geometry, with profound implications for theoretical physics, particularly string theory and M-theory. A central and notoriously difficult question in this field is the compactness problem: identifying, classifying, and constructing all compact Riemannian manifolds whose holonomy group is a proper subgroup of SO(n). Such manifolds possess an abundance of parallel geometric structures, leading to exceptional properties and significant constraints on their topology and geometry. This paper provides a comprehensive review of the compactness problem, exploring its theoretical foundations, existing solutions, and persistent challenges. We delve into the Berger's classification of possible special holonomy groups and examine specific cases such as Calabi-Yau manifolds (holonomy SU(m)), G$_2$ manifolds, and Spin(7) manifolds. The discussion encompasses the analytical and topological techniques employed in their construction and analysis, including elliptic partial differential equations, variational methods, and deformation theory. We highlight the deep connections between special holonomy, Ricci-flat metrics, and supersymmetry, particularly in the context of the string landscape. The paper also addresses the crucial role of parallel forms and spinors in characterizing these geometries and the obstructions to their existence on compact manifolds. Finally, we outline key open problems and future research directions, underscoring the enduring significance of the compactness problem in modern geometry and physics. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17834139 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Compactness Problem for Special Holonomy Manifolds Revista, Zen MATH, 10 The study of special holonomy manifolds represents a vibrant and challenging area within differential geometry, with profound implications for theoretical physics, particularly string theory and M-theory. A central and notoriously difficult question in this field is the compactness problem: identifying, classifying, and constructing all compact Riemannian manifolds whose holonomy group is a proper subgroup of SO(n). Such manifolds possess an abundance of parallel geometric structures, leading to exceptional properties and significant constraints on their topology and geometry. This paper provides a comprehensive review of the compactness problem, exploring its theoretical foundations, existing solutions, and persistent challenges. We delve into the Berger's classification of possible special holonomy groups and examine specific cases such as Calabi-Yau manifolds (holonomy SU(m)), G$_2$ manifolds, and Spin(7) manifolds. The discussion encompasses the analytical and topological techniques employed in their construction and analysis, including elliptic partial differential equations, variational methods, and deformation theory. We highlight the deep connections between special holonomy, Ricci-flat metrics, and supersymmetry, particularly in the context of the string landscape. The paper also addresses the crucial role of parallel forms and spinors in characterizing these geometries and the obstructions to their existence on compact manifolds. Finally, we outline key open problems and future research directions, underscoring the enduring significance of the compactness problem in modern geometry and physics. |
| title | The Compactness Problem for Special Holonomy Manifolds |
| url | https://doi.org/10.5281/zenodo.17834139 |