Domenico Fiorenza: Integrals detecting degree 3 string cobordism classes

Abstract:

The third string bordism group \(\operatorname{Bord}^{\text{String}}\) 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 \(\operatorname{Bord}^{\text{String}}_3 \to \mathbb{Z}/24\mathbb{Z}\). We will show how these formulas naturally emerge when one considers certain natural \(U(1)\)-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.

Based on joint work with Eugenio Landi; arXiv:2209.12933.