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The caloron correspondence can be understood as an equivalence of categoriesbetween $G$-bundles over circle bundles and $LG \rtimes_\rho S^1$-bundles where$LG$ is the group of smooth loops in $G$. We use it, and lifting bundle gerbes,to derive an explicit differential form based formula for the (real) stringclass of an $LG \rtimes_\rho S^1$-bundle.

Circle actions, central extensions and string structures