String bordism invariants in dimension 3 from U(1)-valued TQFTs

Abstract: The third string bordism group is known to be $\mathbb{Z}/24\mathbb{Z}$. 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 ${\rm Bord}_3^{\rm String} \to \mathbb{Z}/24\mathbb{Z}$ (these formulas have been recently rediscovered by Gaiotto–Johnson-Freyd–Witten). In the talk I will show how these formulas naturally emerge when one considers the U(1)-valued 3d TQFTs associated with the classifying stacks of Spin bundles with connection and of String bundles with geometric structure. Joint work with Eugenio Landi (arXiv:2209.12933).

mathematical physicsalgebraic topologycategory theorygeometric topologyquantum algebra

Audience: researchers in the topic

Topology and Geometry Seminar (Texas, Kansas)