The Poincare Lemma for Codifferential, Anticoexact Forms, and Applications to Physics
| Authors | |
|---|---|
| Year of publication | 2022 |
| Type | Article in Periodical |
| Magazine / Source | Results in Mathematics |
| MU Faculty or unit | |
| Citation | |
| Web | https://doi.org/10.1007/s00025-022-01646-z |
| Doi | http://dx.doi.org/10.1007/s00025-022-01646-z |
| Keywords | Poincare lemma; Codifferential; Anticoexact differential forms; Homotopy operator; Clifford bundle; Maxwell equations; Dirac operator; Kalb-Ramond equations; de Rham theory |
| Description | The linear homotopy theory for codifferential operator on Riemannian manifolds is developed in analogy to a similar idea for exterior derivative. The main object is the cohomotopy operator, which singles out a module of anticoexact forms from the module of differential forms defined on a star-shaped open subset of a manifold. It is shown that there is a direct sum decomposition of a differential form into coexact and anticoexat parts. This decomposition gives a new way of solving exterior differential systems. The method is applied to equations of fundamental physics, including vacuum Dirac-Kahler equation, coupled Maxwell-Kalb-Ramond system of equations occurring in a bosonic string theory and its reduction to the Dirac equation. |
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