  In the present work we provide a constructive method to describe contact
structures on compact homogeneous contact manifolds. The main feature of our
approach is to describe the Cartan-Ehresmann connection (gauge field) for
principal circle bundles over complex flag manifolds by using elements of
representation theory of simple Lie algebras. This description allows us to
compute explicitly the expression of the contact form for any Boothby-Wang
fibration over complex flag manifolds as well as their induced homogeneous
Sasaki-Einstein structures. As an application of our results we use the
Cartan-Remmert reduction and the Calabi ansatz technique to provide many
explicit examples of crepant resolutions of Calabi-Yau cones with certain
homogeneous Sasaki-Einstein manifolds realized as links of isolated
hypersurface singularities.
