A Fourier-Mukai transform for real torus bundles
Abstract
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 $\cS$ 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,∇)
(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