Nahm Poles and 0-Instantons
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915862527606784 |
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| author | Usula, Marco |
| author_facet | Usula, Marco |
| contents | We study self-dual 0-connections, or 0-instantons, on asymptotically hyperbolic 4-manifolds. These connections develop a uniform singularity along the conformal infinity, and are asymptotic, at each point of the boundary, to a "Nahm pole" model solution on $H^4$. Examples include the Levi-Civita spin connections on $S^+$ over spin Poincaré-Einstein 4-manifolds. Inspired by the Fefferman-Graham expansion for Poincaré-Einstein metrics, we study the asymptotic expansion of these 0-instantons. We prove that the expansion is log-smooth, and that the coefficient of the first log term - which we call the 0-instanton obstruction tensor - is a conformal invariant related to the Weyl curvature of the ambient conformal metric. We then show that this invariant vanishes if and only if the 0-instanton is smooth modulo gauge. Finally, we study the renormalized Yang-Mills energy: we prove that, if the metric is asymptotically Poincaré-Einstein to third order, then this energy is a well-defined conformal invariant, and equals the negative Chern-Simons invariant of the conformal infinity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_13805 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Nahm Poles and 0-Instantons Usula, Marco Differential Geometry Mathematical Physics Analysis of PDEs 53-00, 53C07, 58J32, 35J60 We study self-dual 0-connections, or 0-instantons, on asymptotically hyperbolic 4-manifolds. These connections develop a uniform singularity along the conformal infinity, and are asymptotic, at each point of the boundary, to a "Nahm pole" model solution on $H^4$. Examples include the Levi-Civita spin connections on $S^+$ over spin Poincaré-Einstein 4-manifolds. Inspired by the Fefferman-Graham expansion for Poincaré-Einstein metrics, we study the asymptotic expansion of these 0-instantons. We prove that the expansion is log-smooth, and that the coefficient of the first log term - which we call the 0-instanton obstruction tensor - is a conformal invariant related to the Weyl curvature of the ambient conformal metric. We then show that this invariant vanishes if and only if the 0-instanton is smooth modulo gauge. Finally, we study the renormalized Yang-Mills energy: we prove that, if the metric is asymptotically Poincaré-Einstein to third order, then this energy is a well-defined conformal invariant, and equals the negative Chern-Simons invariant of the conformal infinity. |
| title | Nahm Poles and 0-Instantons |
| topic | Differential Geometry Mathematical Physics Analysis of PDEs 53-00, 53C07, 58J32, 35J60 |
| url | https://arxiv.org/abs/2603.13805 |