Floer sections in multisymplectic geometry
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918233973456896 |
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| author | Brilleslijper, Ronen Fabert, Oliver |
| author_facet | Brilleslijper, Ronen Fabert, Oliver |
| contents | In symplectic geometry, Floer theory is the most important tool to prove the existence of time-periodic solutions in Hamiltonian mechanics. The core observation is that the $L^2$-gradient lines of the symplectic action functional are pseudo-holomorphic curves, enabling the use of elliptic PDE methods. Multisymplectic geometry is the geometric framework underlying Hamiltonian field theory, where the time line is replaced by higher-dimensional manifolds. In the case of two dimensions and using complex structures, we introduce a novel multisymplectic framework that is fit for the generalization of the elliptic methods from symplectic geometry. Besides proving a Darboux theorem, we show that the $L^2$-gradient lines of our multisymplectic action functional are now pseudo-Fueter curves defined using a compatible almost hyperkähler structure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_05797 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Floer sections in multisymplectic geometry Brilleslijper, Ronen Fabert, Oliver Symplectic Geometry Mathematical Physics Differential Geometry 35J60, 37J39, 53D40, 53Z05, 70S20 In symplectic geometry, Floer theory is the most important tool to prove the existence of time-periodic solutions in Hamiltonian mechanics. The core observation is that the $L^2$-gradient lines of the symplectic action functional are pseudo-holomorphic curves, enabling the use of elliptic PDE methods. Multisymplectic geometry is the geometric framework underlying Hamiltonian field theory, where the time line is replaced by higher-dimensional manifolds. In the case of two dimensions and using complex structures, we introduce a novel multisymplectic framework that is fit for the generalization of the elliptic methods from symplectic geometry. Besides proving a Darboux theorem, we show that the $L^2$-gradient lines of our multisymplectic action functional are now pseudo-Fueter curves defined using a compatible almost hyperkähler structure. |
| title | Floer sections in multisymplectic geometry |
| topic | Symplectic Geometry Mathematical Physics Differential Geometry 35J60, 37J39, 53D40, 53Z05, 70S20 |
| url | https://arxiv.org/abs/2512.05797 |