Secondary Characteristic Classes in CR Geometry

We construct a family of $\mathbb{R}\mathrm{/}\mathbb{Z}$-valued CR invariants for compact strictly pseudoconvex CR manifolds admitting pseudo-Einstein contact forms. We define these invariants by applying the theory of Cheeger–Simons differential characters to a globally defined modification of the normal tractor connection. When the CR holomorphic tangent bundle is trivial, their natural $\mathbb{R}$-valued lifts agree with the generalized Burns–Epstein invariants. For CR manifolds arising as boundaries of relatively compact strictly pseudoconvex domains, we identify these differential characters with those determined by the renormalized connection and derive bulk–boundary formulas involving renormalized characteristic numbers and residues of Chern classes. These formulas recover and extend the results of Burns–Epstein and Marugame. We also obtain obstructions to CR embeddings into complex Euclidean space.