Global Fukaya category II: applications

To paraphrase, part I constructs a bundle of $A_{\mathrm{\infty }}$ categories given the input of a Hamiltonian fibration over a smooth manifold. Here we show that this bundle is generally non-trivial by a sample computation. One principal application is differential geometric, and the other is about algebraic $K$-theory of the integers and the rationals. We find new curvature constraint phenomena for smooth and singular $\mathcal{G}$-connections on principal $\mathcal{G}$-bundles over $S^4$, where $\mathcal{G}$ is $\mathrm{PU}(2)$ or $\mathrm{Ham}(S^2)$. Even for the classical group $\mathrm{PU}(2)$ these phenomena are inaccessible to known techniques like the Yang-Mills theory. The above mentioned computation is the geometric component used to show that the categorified algebraic $K$-theory of the integers and the rationals, defined in ~\cite{cite_SavelyevAlgKtheory} following Toën, admits a $\mathbb{Z}$ injection in degree $4$. This gives a path from Floer theory to number theory.