  The caloron correspondence is a tool that gives an equivalence between
principal $G$-bundles based over the manifold $M \times S^1$ and principal
$LG$-bundles on $M$, where $LG$ is the Fr\'echet Lie group of smooth loops in
the Lie group $G$. This thesis uses the caloron correspondence to construct
certain differential forms called "string potentials" that play the same role
as Chern-Simons forms for loop group bundles. Following their construction, the
string potentials are used to define degree 1 differential characteristic
classes for $\Omega U(n)$-bundles.
  The notion of an "$\Omega$ vector bundle" is introduced and a caloron
correspondence is developed for these objects. Finally, string potentials and
$\Omega$ vector bundles are used to define an $\Omega$ bundle version of the
structured vector bundles of Simons--Sullivan. The "$\Omega$ model" of odd
differential $K$-theory is constructed using these objects and an elementary
differential extension of odd $K$-theory due to Tradler et al.
