Topoi for Photons:A Categorical Formulation of Electromagnetic Duality
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2025
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| author | de ceuster, peter |
| author_facet | de ceuster, peter |
| contents | <p>We propose a categorical formulation of electromagnetic (Maxwell) duality by identifying a class of Grothendieck topoi (and suitable stacky sheaf models) whose internal cohomological data encode electric and magnetic sectors and admit a canonical duality isomorphism. Understanding this duality should ultimately improve our success-rate studying unobserved photonic laws. Building on the program of topospotentials and <span><span><span><span><span>G</span></span></span></span></span>-Theory–Maxwell correspondences, we formulate the Toposic Maxwell Duality Conjecture which asserts a natural equivalence between internal hypercohomology functors associated to dual gauge stacks. We provide precise definitions of the topos model <span><span><span><span><span>E</span></span></span></span></span> over a smooth spacetime manifold <span><span><span><span><span>M</span></span></span></span></span>, the gauge stack <span><span><span><span><span>G</span></span></span></span></span> encoding <span><span><span><span><span>U</span><span>(</span><span>1</span><span>)</span></span></span></span></span>-connections (and higher analogues), and the pair of functors <span><span><span><span><span>F</span><span>,</span><span>G</span></span></span></span></span> selecting electric/magnetic sectors. We prove a partial result: on compact orientable surfaces (notably <span><span><span><span><span>M</span><span>=</span></span><span><span><span>T</span><span><span><span><span><span><span><span>2</span></span></span></span></span></span></span></span></span></span></span></span>) the conjectured duality reduces to Poincaré/Čech–de Rham duality and can be established up to the expected torsion and orientation twists. A worked toy computation on the two-torus exhibits the isomorphism of the relevant cohomological invariants. We conclude with numerical/heuristic checks (lattice discretizations and spectral invariants), physical implications for photonic systems, and explicit open problems, though we affirm the development of experiments, a variety of checks and the debate related to potential implications needs to broaden.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17352601 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Topoi for Photons:A Categorical Formulation of Electromagnetic Duality de ceuster, peter forPhotons Topoi duality electromagnetism Electromagnetism <p>We propose a categorical formulation of electromagnetic (Maxwell) duality by identifying a class of Grothendieck topoi (and suitable stacky sheaf models) whose internal cohomological data encode electric and magnetic sectors and admit a canonical duality isomorphism. Understanding this duality should ultimately improve our success-rate studying unobserved photonic laws. Building on the program of topospotentials and <span><span><span><span><span>G</span></span></span></span></span>-Theory–Maxwell correspondences, we formulate the Toposic Maxwell Duality Conjecture which asserts a natural equivalence between internal hypercohomology functors associated to dual gauge stacks. We provide precise definitions of the topos model <span><span><span><span><span>E</span></span></span></span></span> over a smooth spacetime manifold <span><span><span><span><span>M</span></span></span></span></span>, the gauge stack <span><span><span><span><span>G</span></span></span></span></span> encoding <span><span><span><span><span>U</span><span>(</span><span>1</span><span>)</span></span></span></span></span>-connections (and higher analogues), and the pair of functors <span><span><span><span><span>F</span><span>,</span><span>G</span></span></span></span></span> selecting electric/magnetic sectors. We prove a partial result: on compact orientable surfaces (notably <span><span><span><span><span>M</span><span>=</span></span><span><span><span>T</span><span><span><span><span><span><span><span>2</span></span></span></span></span></span></span></span></span></span></span></span>) the conjectured duality reduces to Poincaré/Čech–de Rham duality and can be established up to the expected torsion and orientation twists. A worked toy computation on the two-torus exhibits the isomorphism of the relevant cohomological invariants. We conclude with numerical/heuristic checks (lattice discretizations and spectral invariants), physical implications for photonic systems, and explicit open problems, though we affirm the development of experiments, a variety of checks and the debate related to potential implications needs to broaden.</p> |
| title | Topoi for Photons:A Categorical Formulation of Electromagnetic Duality |
| topic | forPhotons Topoi duality electromagnetism Electromagnetism |
| url | https://doi.org/10.5281/zenodo.17352601 |