Integrals detecting degree 3 string cobordism classes

The third string bordism group ${\mathrm{B}\mathrm{o}\mathrm{r}\mathrm{d}}_3^{\mathrm{S}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}}$ is known to be $\mathbb{Z}\mathrm{/}24\mathbb{Z}$. Using Waldorf’s notion of a geometric string structure on a manifold, Bunke–Naumann and Redden have exhibited integral formulas involving the Chern-Weil form representative of the first Pontryagin class and the canonical 3-form of a geometric string structure that realize the isomorphism ${\mathrm{B}\mathrm{o}\mathrm{r}\mathrm{d}}_3^{\mathrm{S}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}}\to \mathbb{Z}\mathrm{/}24\mathbb{Z}$. We will show how these formulas naturally emerge when one considers certain natural $\mathrm{U}(1)$-valued and $\mathbb{R}$-valued 3d TQFT associated with the classifying stacks of Spin bundles with connection and of String bundles with geometric structure, respectively.