A Fourier-Mukai transform for real torus bundles

We construct a Fourier--Mukai transform for smooth complex vector bundles $E$ over a torus bundle $\pi:M \to B,$ 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 $\cS$ supported on a spectral covering $\Sigma \hra \what M,$ where $\hat\pi:\what{M} \to B$ is the flat dual fiber bundle. Additional structures on $(E,\nabla)$ (flatness, anti-self-duality) will be reflected by corresponding data on the transform $(\cS, \Sigma).$ Several variations of this construction will be presented, emphasizing the aspects of foliation theory which enter into this picture