Hamiltonian elements in algebraic K-theory

A Hamiltonian bundle $M \hookrightarrow P \to X$ (with monotone compact fibers) induces via Floer theory a type of ``bundle of $A _{\infty}$ categories'' over $X$, with fiber given by the Fukaya category of $M$. Morita theory of $A _{\infty}$ categories, the above picture for $X=S ^{m}$, and geometric representation theory yield the following: if $G$ is a compact Lie group and $R$ is a commutative ring then there is a natural group homomorphism $\pi _{m} (BG) \to K ^{Cat}_{m}(R)$, where $K ^{Cat} _{m} (R)$ are a type of categorified algebraic $K$-theory groups of $R$, analogous to To\"en's secondary $K$-theory. We also construct underlying maps of this type to classical algebraic $K$-theory of $R$. This framework gives a geometry-powered proof that $K ^{Cat} _{2} (\mathbb{Z} )$ is infinitely generated (with the details to appear in a future work). This is in contrast to Quillen's finite generation result for standard algebraic $K$-theory of $\mathbb{Z}$. Taking the Langlands dual of $G$, we explore a conjectural relationship between the images of the corresponding homomorphisms above.