Integrals detecting degree 3 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.

25 pages, exposition improved, references added. Final version

References in corpus (4)

• Twisted differential String and Fivebrane structures
• Čech cocycles for differential characteristic classes -- An infinity-Lie theoretic construction
• The E8 moduli 3-stack of the C-field in M-theory
• String structures and canonical 3-forms