nLab
integral Wu structure

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Defintion

A differential integral Wu structure in degree 2k2k on an oriented smooth manifold XX is a refinement of the Wu class ν 2kH 2k(X, 2)\nu_{2k} \in H^{2k}(X, \mathbb{Z}_2) ν 2k\nu_{2k} by a cocycle ϕ\phi in degree 2k2k ordinary differential cohomology H diff 2k(X)H^{2k}_{diff}(X), hence a circle (2k-1)-bundle with connection 2k1\nabla_{2k-1} whose underlying higher Dixmier-Douady class DD( 2k1)DD(\nabla_{2k-1}) equals ν 2k\nu_{2k} modulo 2-reduction

DD( 2k1)mod2=ν 2kH 2k(X, 2). 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.

manifold dimensioninvariantquadratic formquadratic refinement
4k4ksignature genusintersection pairingintegral Wu structure
4k+24k+2Kervaire invariantframing

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

line bundlesquare rootchoice corresponds to
canonical bundleTheta characteristicover Riemann surface and Hermitian manifold (e.g.Kähler manifold): spin structure
density bundlehalf-density bundle
canonical bundle of Lagrangian submanifoldmetalinear structuremetaplectic correction
determinant line bundlePfaffian line bundle
quadratic secondary intersection pairingpartition function of self-dual higher gauge theoryintegral Wu structure

References

The notion was introduced in def. 2.12 of

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

A smooth stack refinement is considered in

Revised on September 10, 2014 04:33:32 by Urs Schreiber (185.26.182.29)