Structured bundles define differential K-theory. (English) Zbl 1204.57021

Hijazi, Oussama (ed.), Géométrie différentielle, physique mathématique, mathématiques et société (I). Volume en l’honneur de Jean Pierre Bourguignon. Paris: Société Mathématique de France (ISBN 978-2-85629-258-7/pbk). Astérisque 321, 1-3 (2008).
Summary: Complex bundles with connection up to isomorphism form a semigroup under Whitney sum which is far from being a group. We define a new equivalence relation (structured equivalence) so that stable isomorphism classes up to structured equivalence form a group which is describable in terms of the Chern character form plus some finite type invariants from algebraic topology. The elements in this group also satisfy two further somewhat contradictory properties: a locality or gluing property and an integrality property. There is interest in using these objects as prequantum fields in gauge theory and \(M\)-theory.
For the entire collection see [Zbl 1160.53003].


57N65 Algebraic topology of manifolds
55N15 Topological \(K\)-theory
19E08 \(K\)-theory of schemes
58J28 Eta-invariants, Chern-Simons invariants
14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)
57R19 Algebraic topology on manifolds and differential topology