The universal gerbe, Dixmier-Douady class, and gauge theory

We clarity the relation between the Dixmier-Douady class on the space of self-adjoint Fredholm operators ("universal B-field") and the curvature of determinant bundles over infinite-dimensional Grassmannians. In particular, in the case of Dirac type operators on a three dimensional compact manifold we obtain a simple and explicit expression for both forms.