Geometry of the non-Abelian 2-index potential and twisted de Rham cohomology

It is found that the 2-index potential in nonabelian theories does not behave geometrically as a connection but that, considered as an element of the second de Rham cohomology group twisted by a flat connection, it fits well with all the properties assigned to it in various physical contexts. We also prove some results on the Euler characteristic of the twisted de Rham complex. Finally, provided that some conditions are satisfied, we propose a non-Abelian generalisation of S-duality.