Homogeneous Contact Manifolds and Resolutions of Calabi–Yau Cones

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 U(1)-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 (Boothby and Wang in Ann Math 68:721–734, 1958) as well as their underlying Sasaki structures. By following Conlon and Hein (Duke Math J 162:2855–2902, 2013), Van Coevering (Math Ann, 2009. https://doi.org/10.1007/s00208-009-0446-1) and Goto (J Math Soc Jpn 64:1005–1052, 2012), as an application of our results we use the Cartan–Remmert reduction (Grauert in Math Ann 146:331–368, 1962) and the Calabi Ansatz technique (Calabi in Ann Sci École Norm Sup (4) 12:269–294, 1979) to provide many explicit examples of crepant resolutions of Calabi–Yau cones with certain homogeneous Sasaki–Einstein manifolds realized as links of isolated singularities. These concrete examples illustrate the existence part of the conjecture introduced in Martelli and Sparks (Phys Rev D 79(6):065009, 2009).