The Poincare Lemma for Codifferential, Anticoexact Forms, and Applications to Physics

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KYCIA Radoslaw Antoni

Year of publication 2022
Type Article in Periodical
Magazine / Source Results in Mathematics
MU Faculty or unit

Faculty of Science

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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