Journal of Geometry and Physics 50 (2004) 360–392 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 bundlesE 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 S supported on a spectral covering Σ ↪ → M̂, where π ̂ : M ̂ → B is the flat dual fiber bundle. Additional structures on (E,∇) (flatness, anti-self-duality) will be reflected by corresponding data on the transform (S,Σ). Several variations of this construction will be presented, emphasizing the aspects of foliation theory which enter into this picture