Non-Split Geometry on Products of Vector Bundles

We propose a model in which a spliced vector bundle (with an arbitrary number of gauge structures in the splice) possesses a geometry which do not split. The model employs connection 1-forms with values in a space-product of Lie algebras, and therefore interlaces the various gauge structures in a non-trivial manner. Special attention is given to the structure of the geometric ghost sector and the super-algebra it possesses: The ghosts emerge as \(x\)-dependent deformations at the gauge sector, and the associated BRST super algebra is realized as constraints that follow from the invariance of the curvature.