A Fourier–Mukai transform for real torus bundles

We construct a Fourier–Mukai transform for smooth complex vector bundles E over a torus bundle π:M→B, the vector bundles being endowed with various structures of increasing complexity. At a minimum, we consider vector bundles E with a flat partial unitary connection, that is families or deformations of flat vector bundles (or unitary local systems) on the torus T. This leads to a correspondence between such objects on M and relative skyscraper sheaves supported on a spectral covering , where is the flat dual fiber bundle. Additional structures on (E,) (flatness, anti-self-duality) will be reflected by corresponding data on the transform . Several variations of this construction will be presented, emphasizing the aspects of foliation theory which enter into this picture.