De Rham-Kodaira's Theorem and Dual Gauge Transformations

A general action is proposed for the fields of $q$-dimensional differential form over the compact Riemannian manifold of arbitrary dimensions. Mathematical tools are based on the well-known de Rham-Kodaira decomposing theorem on harmonic integral. A field-theoretic action for strings, $p$-branes and high-spin fields is naturally derived. We also have, naturally, the generalized Maxwell equations with an electromagnetic and monopole current on a curved space-time. A new type of gauge transformations ({\it dual} gauge transformations) plays an essential role for coboundary $q$-forms.