Saved in:
| Hovedforfatter: | |
|---|---|
| Format: | Recurso digital |
| Sprog: | |
| Udgivet: |
Zenodo
2025
|
| Fag: | |
| Online adgang: | https://doi.org/10.5281/zenodo.17352601 |
| Tags: |
Tilføj Tag
Ingen Tags, Vær først til at tagge denne postø!
|
Indholdsfortegnelse:
- <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>