nLab 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 $\phi$ in degree $2k$ ordinary differential cohomology $H^{2k}_{diff}(X)$, hence a circle (2k-1)-bundle with connection $\nabla_{2k-1}$ whose underlying higher Dixmier-Douady class $DD(\nabla_{2k-1})$ equals $\nu_{2k}$ modulo 2-reduction

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

These are the characteristic elements of the intersection product on ordinary cohomology/ordinary differential cohomology, inducing its quadratic refinements.

Related concepts

• Wu class, twisted Wu structure

• canonical bundle

• Spin Chern-Simons theory

• integral Steenrod square

• integral Stiefel-Whitney class

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 and Hermitian manifold (e.g.Kähler manifold): 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