Hamiltonian elements in algebraic K-theory

Recall that topological complex K-theory associates to an isomorphism class of a complex vector bundle E over a space X an element of the complex K-theory group of X. Or from algebraic K-theory perspective, one assigns a homotopy class X K (K), where K is the ring of compact operators on the Hilbert space. We show that there is an analogous story for algebraic K-theory of a general commutative ring k, replacing complex vector bundles by certain Hamiltonian fiber bundles. The construction actually first assigns elements in a certain categorified algebraic K-theory, analogous to To\"en's secondary K-theory of k. And there is a natural map from this categorified algebraic K-theory to the classical variant.