Hamiltonian elements in algebraic K-theory

Abstract

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 $[X \to K (\mathcal{K})]$, where $\mathcal{K}$ 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.