Conformally compact and higher conformal Yang-Mills equations
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| Format: | Preprint |
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2023
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| _version_ | 1866910367535333376 |
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| author | Gover, A. Rod Latini, Emanuele Waldron, Andrew Zhang, Yongbing |
| author_facet | Gover, A. Rod Latini, Emanuele Waldron, Andrew Zhang, Yongbing |
| contents | On conformally compact manifolds we study Yang-Mills equations, their boundary conditions, formal asymptotics, and Dirichlet-to-Neumann maps. We find that smooth solutions with "magnetic" Dirichlet boundary data are obstructed by a conformally invariant, higher order boundary current. We study the asymptotics of the interior Yang-Mills energy functional and show that the obstructing current is the variation of the conformally invariant coefficient of the first log term in this expansion which is a higher derivative conformally invariant analog of the Yang-Mills energy. The invariant energy is the anomaly for the renormalized interior Yang-Mills functional and its variation gives higher conformal Yang-Mills equations. Global solutions to the magnetic boundary problem determine higher order "electric" Neumann data. This yields the Dirichlet-to-Neumann map. We also construct conformally invariant, higher transverse derivative boundary operators. Acting on interior connections, they give obstructions to solving the Yang-Mills boundary problem, determine the asymptotics of formal solutions, and yield conformally invariant tensors capturing the (non-local) electric Neumann data. We also characterize a renormalized Yang-Mills action functional that encodes global features analogously to the renormalized volume for Poincaré-Einstein structures. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_11458 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Conformally compact and higher conformal Yang-Mills equations Gover, A. Rod Latini, Emanuele Waldron, Andrew Zhang, Yongbing Differential Geometry General Relativity and Quantum Cosmology High Energy Physics - Theory Mathematical Physics On conformally compact manifolds we study Yang-Mills equations, their boundary conditions, formal asymptotics, and Dirichlet-to-Neumann maps. We find that smooth solutions with "magnetic" Dirichlet boundary data are obstructed by a conformally invariant, higher order boundary current. We study the asymptotics of the interior Yang-Mills energy functional and show that the obstructing current is the variation of the conformally invariant coefficient of the first log term in this expansion which is a higher derivative conformally invariant analog of the Yang-Mills energy. The invariant energy is the anomaly for the renormalized interior Yang-Mills functional and its variation gives higher conformal Yang-Mills equations. Global solutions to the magnetic boundary problem determine higher order "electric" Neumann data. This yields the Dirichlet-to-Neumann map. We also construct conformally invariant, higher transverse derivative boundary operators. Acting on interior connections, they give obstructions to solving the Yang-Mills boundary problem, determine the asymptotics of formal solutions, and yield conformally invariant tensors capturing the (non-local) electric Neumann data. We also characterize a renormalized Yang-Mills action functional that encodes global features analogously to the renormalized volume for Poincaré-Einstein structures. |
| title | Conformally compact and higher conformal Yang-Mills equations |
| topic | Differential Geometry General Relativity and Quantum Cosmology High Energy Physics - Theory Mathematical Physics |
| url | https://arxiv.org/abs/2311.11458 |