  In the first part of this paper, given a smooth family of Dirac-type
operators on an odd-dimensional closed manifold, we construct an abelian
gerbe-with-connection whose curvature is the three-form component of the
Atiyah-Singer families index theorem. In the second part of the paper, given a
smooth family of Dirac-type operators whose index lies in the i-th filtration
of the reduced K-theory of the parametrizing space, we construct a set of
Deligne cohomology class of degree i whose curvatures are the i-form component
of the Atiyah-Singer families index theorem.

Differential Geometry