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integral Wu structure

Defintion

A differential integral Wu structure in degree $2k$ on an oriented smooth manifold $X$ is a refinement of the Wu class ${\nu }_{2k}\in H^{2k}(X,{\mathbb{Z}}_2)$ ${\nu }_{2k}$ by a cocycle $\varphi$ in degree $2k$ ordinary differential cohomology $H_{\mathrm{diff}}^{2k}(X)$, hence a circle (2k-1)-bundle with connection ${\nabla }_{2k-1}$ whose underlying higher Dixmier-Douady class $\mathrm{DD}({\nabla }_{2k-1})$ equals ${\nu }_{2k}$ modulo 2-reduction

$$\mathrm{DD}({\nabla }_{2k-1})\mathrm{mod}2={\nu }_{2k}\in H^{2k}(X,{\mathbb{Z}}_2).$$DD(\nabla_{2k-1}) mod 2 = \nu_{2k} \in H^{2k}(X, \mathbb{Z}_2) \,.

Related concepts

• Wu class, twisted Wu structure

• canonical bundle

manifold dimension	invariant	quadratic form	quadratic refinement
$4k$	signature genus	intersection pairing	integral Wu structure
$4k+2$	Kervaire invariant		framing

The following table lists classes of examples of square roots of line bundles

line bundle	square root	choice corresponds to
canonical bundle	Theta characteristic	over Riemann surface: spin structure
density bundle	half-density bundle
canonical bundle of Lagrangian submanifold	metalinear structure	metaplectic correction
determinant line bundle	Pfaffian line bundle
quadratic secondary intersection pairing	partition function of self-dual higher gauge theory	integral Wu structure

References

The notion was introduced in def. 2.12 of

• Mike Hopkins, Isadore Singer, Quadratic Functions in Geometry, Topology, and M-Theory

motivated by considerations about abelian 7d Chern-Simons theory in

• Edward Witten, Five-Brane Effective Action In M-Theory J. Geom. Phys.22:103-133,1997 (arXiv:hep-th/9610234)

• Edward Witten, On flux quantization in M-theory and the effective action (arXiv:hep-th/9609122v2)

A smooth stack refinement is considered in

• Domenico Fiorenza, Hisham Sati, Urs Schreiber The moduli 3-stack of the C-field