Integrals detecting degree 3 string cobordism classes

The third string bordism group Bord^String_3 is known to be Z/24Z. 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 Bord^String_3 → Z/24Z. We will show how these formulas naturally emerge when one considers certain natural U(1)-valued and R-valued 3d TQFT associated with the classifying stacks of Spin bundles with connection and of String bundles with geometric structure, respectively.