The Caloron Correspondence and Odd Differential K-theory

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.