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 $T\mathrm{.}$ This leads to a correspondence between such objects on $M$ and relative skyscraper sheaves $\text{cS}$ 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 $(E,\mathrm{\nabla })$ (flatness, anti-self-duality) will be reflected by corresponding data on the transform $(\text{cS},\mathrm{\Sigma })\mathrm{.}$ Several variations of this construction will be presented, emphasizing the aspects of foliation theory which enter into this picture