Classical Dixmier-Douady theory gives a full classification of C*-algebra bundles with compact operators as fibres by the third cohomology group of the underlying space. In joint work with Marius Dadarlat we showed that this and other results by Dixmier and Douady can be generalized to bundles with fibers isomorphic to stabilized strongly self-absorbing C*-algebras. In this talk I will point out a construction that is joint work with David Evans and produced interesting examples of such bundles over the Lie groups SU(n) constructed from exponential functors. They arise naturally as Fell bundles and are equivariant with respect to the conjugation action of the group on itself. For the determinant functor our construction reproduces the basic gerbe over SU(n) and the equivariant K-theory of its section algebra is isomorphic to the Verlinde ring by results of Freed, Hopkins and Teleman. In general we end up with bundles representing higher twists of K-theory.