  In this paper, we construct the index bundle gerbe of a family of
self-adjoint Dirac-type operators, refining a construction of Segal. In a
special case, we construct a geometric bundle gerbe called the caloron bundle
gerbe, which comes with a natural connection and curving, and show that it is
isomorphic to the analytically constructed index bundle gerbe. We apply these
constructions to certain moduli spaces associated to compact Riemann surfaces,
constructing on these moduli spaces, natural bundle gerbes with connection and
curving, whose 3-curvature represent Dixmier-Douady classes that are generators
of the third de Rham cohomology groups of these moduli spaces.
