Integrals detecting degree 3 string cobordism classes

TL;DR

This paper explores integral formulas related to the third string bordism group, connecting geometric string structures, characteristic classes, and topological quantum field theories to understand cobordism classes.

Contribution

It demonstrates how integral formulas for string cobordism classes naturally arise from 3D TQFTs associated with Spin and String bundles with geometric structures.

Findings

Integral formulas realize the isomorphism to  24a0a5a0Z

Formulas involve Chern-Weil forms and 3-forms of geometric string structures

Connection between TQFTs and string cobordism classes

Abstract

The third string bordism group $\mathrm{Bord}_3^{\mathrm{String}}$ 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 $\mathrm{Bord}_3^{\mathrm{String}} \to \mathbb{Z}/24\mathbb{Z}$. 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.